// Numbas version: exam_results_page_options
{"questions": [], "duration": 0, "name": "Matrices", "showQuestionGroupNames": false, "allQuestions": true, "percentPass": 0, "feedback": {"showanswerstate": true, "advicethreshold": 0, "showactualmark": true, "allowrevealanswer": true, "showtotalmark": true}, "shuffleQuestions": false, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find determinant of a 3x3 matrix by row reduction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "d1[1][1]<>0", "maxRuns": 100}, "variables": {"c20": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "c20"}, "c11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)*100", "description": "", "name": "c11"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([m1[0],c11*m1[0]+m1[1],c12*m1[0]+m1[2]])", "description": "", "name": "m2"}, "c12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)*100", "description": "", "name": "c12"}, "upper_triangular": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([\n    d1[0],\n    d1[1],\n    d1[2]-(d1[2][1]/d1[1][1])*d1[1]\n])", "description": "", "name": "upper_triangular"}, "determinant": {"templateType": "anything", "group": "Ungrouped variables", "definition": "det(m1)", "description": "", "name": "determinant"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([\n    m1[0],\n    m1[1]-(m1[1][0]/m1[0][0])*m1[0],\n    m1[2]-(m1[2][0]/m1[0][0])*m1[0]\n])", "description": "", "name": "d1"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(1..4),3),3))", "description": "", "name": "m1"}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix(\n  [m2[0]+c20*m2[1],m2[1],m2[2]]\n)", "description": "", "name": "m3"}}, "ungrouped_variables": ["m1", "c11", "c12", "m2", "c20", "m3", "determinant", "d1", "upper_triangular"], "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "parts": [{"prompt": "<p>Write another matrix of single-digit values (each element of the matrix should be an integer in the range $[-9,9]$) whose determinant is the same as $\\mathrm{M}$.</p>", "markPerCell": false, "allowFractions": false, "correctAnswer": "m1", "showCorrectAnswer": true, "useCustomName": false, "customName": "", "customMarkingAlgorithm": "all_single_digits:\n  if(all(map(isint(studentnumbers[x][y]) and abs(studentnumbers[x][y])<10,[x,y],cell_indexes)),\n    true\n  ,\n    incorrect(\"Your answer is not a matrix of single-digit integer values.\");\n    end();\n    false\n  )\n\ncorrectMatrix: matrix(settings[\"correctAnswer\"])\n\ndifferent:\n  if(studentMatrix<>m3,\n    true\n  ,\n    incorrect(\"You must enter a different matrix to the one given.\");\n    end();\n    false\n  )\n\nright_determinant:\n  correctif(det(studentMatrix)=det(correctMatrix))\n\nmark:\n    apply(any_empty);\n    apply(any_invalid);\n    apply(all_single_digits);\n    apply(different);\n    apply(right_determinant)", "unitTests": [], "correctAnswerFractions": false, "allowResize": false, "numRows": "3", "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": "3", "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}, {"prompt": "<p>Write an upper triangular&nbsp;matrix whose determinant is the same as $\\mathrm{M}$. Enter your answers as integers or fractions, not decimals.</p>", "markPerCell": false, "allowFractions": true, "correctAnswer": "upper_triangular", "showCorrectAnswer": true, "useCustomName": false, "customName": "", "customMarkingAlgorithm": "is_upper_triangular:\n  if(all(map(x<=y or studentmatrix[x][y]=0, [x,y], cell_indexes)),\n    true\n  ,\n    incorrect(\"Your matrix is not upper triangular\");\n    end();\n    false\n  )\n\ncorrectMatrix: matrix(settings[\"correctAnswer\"])\n\nright_determinant:\n  correctif(det(studentMatrix)=det(correctMatrix))\n\nmark:\n    apply(any_empty);\n    apply(any_invalid);\n    apply(is_upper_triangular);\n    apply(right_determinant)", "unitTests": [], "correctAnswerFractions": true, "allowResize": false, "numRows": "3", "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": "3", "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}, {"mustBeReduced": false, "maxValue": "det(m1)", "useCustomName": false, "prompt": "<p>What is the determinant of $\\mathrm{M}$?</p>", "showFractionHint": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "allowFractions": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minValue": "det(m1)", "customName": "", "unitTests": [], "scripts": {}, "showCorrectAnswer": true, "marks": 1, "showFeedbackIcon": true}], "statement": "<p>You are given the matrix $\\mathrm{M} = \\var{m3}$.</p>", "tags": [], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a 3x3 matrix with very big elements, perform row operations to find a&nbsp;matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.</p>"}, "advice": "<h4>a)</h4>\n<p>Note that subtracting one row from another does not affect the determinant. In fact, adding or subtracting any multiple of one row from another&nbsp;does not affect the determinant. That is, the row operation $r_i \\to r_i + ar_j$, $i \\neq j$, does not affect the determinant.</p>\n<p>Subtract $\\simplify{{c20}r_2}$ from $r_1$ to get $\\mathrm{M}_1 = \\var{m2}$.</p>\n<p>Subtract $\\simplify{{c11}r_1}$ from $r_2$ and $\\simplify{{c12}r_1}$ from $r_3$ of $\\mathrm{M}_1$ to get $\\mathrm{M}_2 = \\var{m1}$.</p>\n<h4>b)</h4>\n<p>Considering&nbsp;each column in turn, we&nbsp;use row operations to ensure that there are zeros in each entry below the main diagonal, $\\mathrm{M}_{ij}$, $i \\gt j$.</p>\n<p>Subtract $\\simplify[fractionnumbers,unitfactor]{{m1[1][0]/m1[0][0]}r_1}$ from $r_2$ and&nbsp;$\\simplify[fractionnumbers,unitfactor]{{m1[2][0]/m1[0][0]}r_1}$ from $r_3$ of $\\mathrm{M}_2$ to obtain&nbsp;$\\mathrm{M}_3 = \\simplify[fractionnumbers]{{d1}}$. The first entry of each row, apart from the first row, is zero.</p>\n<p>Subtract $\\simplify[fractionnumbers,unitfactor]{{m1[2][1]/m1[1][1]}r_2}$ from $r_3$ of $\\mathrm{M}_3$ to obtain&nbsp;$\\mathrm{M}_4 = \\simplify[fractionnumbers]{{upper_triangular}}$. The second&nbsp;entry of each row, apart from the second row, is zero.</p>\n<h4>c)</h4>\n<p>Using the upper triangular&nbsp;matrix found in part&nbsp;<strong>b)</strong>, the determinant of $\\mathrm{M}$ is $\\simplify[!basic,fractionnumbers]{{upper_triangular[0][0]}*{upper_triangular[1][1]}*{upper_triangular[2][2]}} = \\var{determinant}$.</p>"}, {"name": "Find determinant of a 3x3 matrix with a free variable by row reduction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "d1[1][1]<>0\nand d1[1][1]*d1[2][1]<0", "maxRuns": 100}, "variables": {"q": {"templateType": "anything", "group": "Constants", "definition": "free_det(upper_triangular, upper_triangular_a)", "description": "", "name": "q"}, "aa": {"templateType": "anything", "group": "Short names for upper triangular matrices", "definition": "upper_triangular_a", "description": "", "name": "aa"}, "ac": {"templateType": "anything", "group": "Short names for upper triangular matrices", "definition": "upper_triangular", "description": "", "name": "ac"}, "c12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)*100", "description": "", "name": "c12"}, "a1": {"templateType": "anything", "group": "A term", "definition": "matrix([\n    [0,0,random(1..4)],\n    [0,0,random(1..4)],\n    [0,0,0]\n])", "description": "", "name": "a1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([\n    m1[0],\n    m1[1]-(m1[1][0]/m1[0][0])*m1[0],\n    m1[2]-(m1[2][0]/m1[0][0])*m1[0]\n])", "description": "", "name": "d1"}, "upper_triangular_a": {"templateType": "anything", "group": "A term", "definition": "matrix([\n    ad1[0],\n    ad1[1],\n    ad1[2]-(d1[2][1]/d1[1][1])*ad1[1]\n])", "description": "", "name": "upper_triangular_a"}, "m1": {"templateType": "anything", "group": "Constants", "definition": "matrix(repeat(repeat(random(1..4),3),3))", "description": "", "name": "m1"}, "c20": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "c20"}, "c11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)*100", "description": "", "name": "c11"}, "m2": {"templateType": "anything", "group": "Constants", "definition": "matrix([m1[0],c11*m1[0]+m1[1],c12*m1[0]+m1[2]])", "description": "", "name": "m2"}, "a3": {"templateType": "anything", "group": "A term", "definition": "matrix(\n  [a2[0]+c20*a2[1],a2[1],a2[2]]\n)", "description": "", "name": "a3"}, "upper_triangular": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([\n    d1[0],\n    d1[1],\n    d1[2]-(d1[2][1]/d1[1][1])*d1[1]\n])", "description": "", "name": "upper_triangular"}, "determinant": {"templateType": "anything", "group": "Ungrouped variables", "definition": "det(m1)", "description": "", "name": "determinant"}, "a2": {"templateType": "anything", "group": "A term", "definition": "matrix([a1[0],c11*a1[0]+a1[1],c12*a1[0]+a1[2]])", "description": "", "name": "a2"}, "ad1": {"templateType": "anything", "group": "A term", "definition": "matrix([\n    a1[0],\n    a1[1]-(m1[1][0]/m1[0][0])*a1[0],\n    a1[2]-(m1[2][0]/m1[0][0])*a1[0]\n])", "description": "", "name": "ad1"}, "m3": {"templateType": "anything", "group": "Constants", "definition": "matrix(\n  [m2[0]+c20*m2[1],m2[1],m2[2]]\n)", "description": "", "name": "m3"}}, "ungrouped_variables": ["c12", "c11", "c20", "determinant", "d1", "upper_triangular"], "functions": {"show_matrix": {"type": "string", "language": "javascript", "definition": "var rows = [];\nfor(var i=0;i<m.rows;i++) {\n  var r = [];\n  for(var j=0;j<m.columns;j++) {\n    r.push('\\\\simplify[all,fractionnumbers]{{'+ma[i][j]+'}*a + {'+m[i][j]+'}}');\n  }\n  rows.push(r.join(' & '));\n}\nvar s = new Numbas.jme.types.TString('\\\\begin{pmatrix} '+rows.join(' \\\\\\\\ ')+' \\\\end{pmatrix}');\ns.latex = true;\nreturn s;", "parameters": [["m", "matrix"], ["ma", "matrix"]]}, "free_det": {"type": "list", "language": "jme", "definition": "[\n  ma[0][0] * ma[1][1] * ma[2][2],\n  ma[0][0] * ma[1][1] * m[2][2] + ma[0][0]*m[1][1] * ma[2][2] + m[0][0] * ma[1][1] * ma[2][2],\n  m[0][0] * m[1][1] * ma[2][2] + m[0][0] * ma[1][1] * m[2][2] + ma[0][0] * m[1][1] * m[2][2],\n  m[0][0] * m[1][1]*m[2][2]\n]", "parameters": [["m", "matrix"], ["ma", "matrix"]]}}, "variable_groups": [{"variables": ["m1", "m2", "m3", "q"], "name": "Constants"}, {"variables": ["a1", "a2", "a3", "ad1", "upper_triangular_a"], "name": "A term"}, {"variables": ["ac", "aa"], "name": "Short names for upper triangular matrices"}], "preamble": {"css": "", "js": ""}, "parts": [{"prompt": "<p>Write a matrix $\\mathrm{M}'$ with the same determinant as $\\mathrm{M}$&nbsp;as the sum of two upper triangular matrices, one of which is multiplied by $a$. Enter numbers as integers or fractions, not decimals.</p>\n<p>$\\mathrm{M}' = $&nbsp;[[0]]&nbsp;$a + \\phantom{}$&nbsp;[[1]]</p>", "showCorrectAnswer": true, "customMarkingAlgorithm": "correctDeterminant: \n  free_det(upper_triangular, upper_triangular_a)\n\nstudentDeterminant:\n  free_det(interpreted_answers[1], interpreted_answers[0])\n\nright_determinant:\n  correctif(all(map(isclose(a,b),[a,b],zip(studentDeterminant,correctDeterminant))))\n\ngap_feedback (Feedback on each of the gaps):\n    map(\n        let(\n            result,submit_part(gaps[gap_number][\"path\"],answer),\n            name,gaps[gap_number][\"name\"],\n                assert(name=\"\",feedback(translate('part.gapfill.feedback header',[\"name\": name])));\n                concat_feedback(filter(x[\"op\"]<>\"warning\",x,result[\"feedback\"]), if(marks>0,result[\"marks\"]/marks,1));\n                result\n        ),\n        [gap_number,answer,index],\n        zip(gap_order,studentAnswer,list(1..len(gaps)))\n    )\n\nboth_valid:\n  map(assert(x[\"valid\"], concat_feedback(x[\"feedback\"],0.5)), x, marked_original_order)\n  \n\nboth_upper_triangular:\n  if(all(map(x[\"credit\"]=1, x, marked_original_order)),\n    true\n  ,\n    incorrect(\"One or both of your matrices are not upper triangular - every value below the main diagonal should be zero.\");\n    end();\n    false\n  )\n\nmark:\n  apply(both_valid);\n  apply(both_upper_triangular);\n  apply(right_determinant)", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "correctAnswer": "upper_triangular_a", "markPerCell": false, "useCustomName": false, "customName": "", "customMarkingAlgorithm": "is_upper_triangular:\n  if(all(map(x<=y or studentmatrix[x][y]=0, [x,y], cell_indexes)),\n    correct();\n    true\n  ,\n    incorrect(\"Your matrix is not upper triangular\");\n    end();\n    false\n  )\n\nmark:\n    apply(any_empty);\n    apply(any_invalid);\n    apply(is_upper_triangular)", "unitTests": [], "correctAnswerFractions": true, "allowResize": false, "numRows": "3", "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": "3", "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "1", "showFeedbackIcon": false}, {"showCorrectAnswer": true, "allowFractions": true, "correctAnswer": "upper_triangular", "markPerCell": false, "useCustomName": false, "customName": "", "customMarkingAlgorithm": "is_upper_triangular:\n  if(all(map(x<=y or studentmatrix[x][y]=0, [x,y], cell_indexes)),\n    correct();\n    true\n  ,\n    incorrect(\"Your matrix is not upper triangular\");\n    end();\n    false\n  )\n\nmark:\n    apply(any_empty);\n    apply(any_invalid);\n    apply(is_upper_triangular)", "unitTests": [], "correctAnswerFractions": true, "allowResize": false, "numRows": "3", "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": "3", "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "1", "showFeedbackIcon": false}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"answer": "{aa[0][0]*aa[1][1]*aa[2][2]}*a^3 + {aa[0][0]*aa[1][1]*ac[2][2] + aa[0][0]*ac[1][1]*aa[2][2] + ac[0][0]*aa[1][1]*aa[2][2]}*a^2 + {ac[0][0]*ac[1][1]*aa[2][2] + ac[0][0]*aa[1][1]*ac[2][2] + aa[0][0]*ac[1][1]*ac[2][2]}*a + {ac[0][0]*ac[1][1]*ac[2][2]}", "vsetRangePoints": 5, "useCustomName": false, "prompt": "<p>What is the determinant of $\\mathrm{M}$, in terms of $a$?</p>", "checkingType": "absdiff", "valuegenerators": [{"value": "", "name": "a"}], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "showCorrectAnswer": true, "marks": 1}], "statement": "<p>You are given the matrix $\\mathrm{M} = <br/>\\begin{pmatrix} <br/>\\simplify{{m3[0][0]} + {a3[0][0]}a} &amp; \\simplify{{m3[0][1]} + {a3[0][1]}a} &amp; \\simplify{{m3[0][2]} + {a3[0][2]}a} \\\\<br/>\\simplify{{m3[1][0]} + {a3[1][0]}a} &amp; \\simplify{{m3[1][1]} + {a3[1][1]}a} &amp; \\simplify{{m3[1][2]} + {a3[1][2]}a} \\\\<br/>\\simplify{{m3[2][0]} + {a3[2][0]}a} &amp; \\simplify{{m3[2][1]} + {a3[2][1]}a} &amp; \\simplify{{m3[2][2]} + {a3[2][2]}a} \\\\<br/>\\end{pmatrix}$ for some $a \\in \\mathbb{R}$.</p>", "tags": [], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a 3x3 matrix with very big elements, perform row operations to find a&nbsp;matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.</p>"}, "advice": "<h4>a)</h4>\n<p>Note that subtracting one row from another does not affect the determinant. In fact, adding or subtracting any multiple of one row from another&nbsp;does not affect the determinant. That is, the row operation $r_i \\to r_i + ar_j$, $i \\neq j$, does not affect the determinant.</p>\n<p>First, we can perform a couple of row operations to make the numbers in each entry smaller.</p>\n<p>Subtract $\\simplify{{c20}r_2}$ from $r_1$ to obtain&nbsp;$\\mathrm{M}_1 = \\var{show_matrix(m2,a2)}$.</p>\n<p>Subtract $\\simplify{{c11}r_1}$ from $r_2$ and $\\simplify{{c12}r_1}$ from $r_3$ of $\\mathrm{M}_1$ to obtain&nbsp;$\\mathrm{M}_2 = \\var{show_matrix(m1,a1)}$.</p>\n<p>Considering&nbsp;each column in turn, we&nbsp;use row operations to ensure that there are zeros in each entry below the main diagonal, $\\mathrm{M}_{ij}$, $i \\gt j$.</p>\n<p>Subtract $\\simplify[fractionnumbers,unitfactor]{{m1[1][0]/m1[0][0]}r_1}$ from $r_2$ and&nbsp;$\\simplify[fractionnumbers,unitfactor]{{m1[2][0]/m1[0][0]}r_1}$ from $r_3$ of $\\mathrm{M}_2$ to obtain&nbsp;$\\mathrm{M}_3 = \\var{show_matrix(d1,ad1)}$.</p>\n<p><span data-jme-visible=\"d1[2][1]*d1[1][1]&lt;0\">Add&nbsp;$\\simplify[fractionnumbers,unitfactor]{{-d1[2][1]/d1[1][1]}r_2}$ to&nbsp;</span><span data-jme-visible=\"d1[2][1]*d1[1][1]&gt;0\">Subtract $\\simplify[fractionnumbers,unitfactor]{{d1[2][1]/d1[1][1]}r_2}$ from</span> $r_3$ of $\\mathrm{M}_3$ to obtain&nbsp;$\\mathrm{M}_4 = \\var{show_matrix(upper_triangular,upper_triangular_a)}$.</p>\n<p>So, $\\mathrm{M}_4$ can be written in the required form as $\\var[fractionnumbers]{upper_triangular_a}a + \\var[fractionnumbers]{upper_triangular}$.</p>\n<h4>b)</h4>\n<p>Using the upper triangular&nbsp;matrix $\\mathrm{M}_4$ found in part&nbsp;<strong>a)</strong>, the determinant of $\\mathrm{M}$ is</p>\n<p></p>\n<p>\\[ \\det(\\mathrm{M}) = \\det(\\mathrm{M}_4) = \\left(\\simplify[all,!noleadingminus,fractionnumbers]{({aa[0][0]}a+{ac[0][0]})}\\right) \\times&nbsp;\\left(\\simplify[all,fractionnumbers]{({aa[1][1]}a+{ac[1][1]})}\\right) \\times&nbsp;\\left(\\simplify[all,fractionnumbers]{({aa[2][2]}a+{ac[2][2]})}\\right) = \\simplify[all,fractionnumbers]{{aa[0][0]*aa[1][1]*aa[2][2]}*a^3 + {aa[0][0]*aa[1][1]*ac[2][2] + aa[0][0]*ac[1][1]*aa[2][2] + ac[0][0]*aa[1][1]*aa[2][2]}*a^2 + {ac[0][0]*ac[1][1]*aa[2][2] + ac[0][0]*aa[1][1]*ac[2][2] + aa[0][0]*ac[1][1]*ac[2][2]}*a + {ac[0][0]*ac[1][1]*ac[2][2]}} \\]</p>"}, {"name": "Find determinants and inverses of 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["a", "b", "c"], "name": "Unnamed group"}], "variables": {"b22": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(tr3*b11=b21*b12,tr3+1,tr3)", "description": "", "name": "b22"}, "b21": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-6..6 except 0)", "description": "", "name": "b21"}, "tr2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "tr2"}, "b": {"group": "Unnamed group", "templateType": "anything", "definition": "matrix([ [b11,b12], [b21,b22] ])", "description": "", "name": "b"}, "c": {"group": "Unnamed group", "templateType": "anything", "definition": "matrix([ [c11,c12], [c21,c22] ])", "description": "", "name": "c"}, "a22": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(tr1*a11=a21*a12,tr1+1,tr1)", "description": "", "name": "a22"}, "c21": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5)", "description": "", "name": "c21"}, "tr3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "tr3"}, "c22": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(tr4*c11=c21*c12,tr4+1,tr4)", "description": "", "name": "c22"}, "c11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,2,4)", "description": "", "name": "c11"}, "tr4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "tr4"}, "b11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a11=tr2,tr2+1,tr2)", "description": "", "name": "b11"}, "tr1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "tr1"}, "a": {"group": "Unnamed group", "templateType": "anything", "definition": "matrix([ [a11,a12],[a21,a22] ])", "description": "", "name": "a"}, "a12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "a12"}, "b12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "b12"}, "c12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a12+b12", "description": "", "name": "c12"}, "a21": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-6..6 except 0)", "description": "", "name": "a21"}, "a11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "a11"}}, "ungrouped_variables": ["a11", "a12", "a21", "a22", "b11", "b12", "b21", "b22", "c11", "c12", "c21", "c22", "tr1", "tr2", "tr3", "tr4"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "functions": {"inverse": {"type": "matrix", "language": "jme", "definition": "matrix([\n    [m[1][1], -m[0][1]],\n    [-m[1][0], m[0][0]]\n])/det(m)", "parameters": [["m", "matrix"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "det(a)", "maxValue": "det(a)", "marks": 0.5}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "det(b)", "maxValue": "det(b)", "marks": 0.5}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "det(c)", "maxValue": "det(c)", "marks": 0.5}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "det(a*b*c)", "maxValue": "det(a*b*c)", "marks": 0.5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Calculate the determinants of these matrices.</p>\n<p>$\\mathrm{det}\\left(A\\right) = $&nbsp;[[0]]</p>\n<p>$\\mathrm{det}\\left(B\\right) = $&nbsp;[[1]]</p>\n<p>$\\mathrm{det}\\left(C\\right) = $&nbsp;[[2]]</p>\n<p>$\\mathrm{det}\\left(ABC\\right) = $&nbsp;[[3]]</p>", "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": true, "correctAnswer": "inverse(a)", "markPerCell": false, "allowResize": false, "correctAnswerFractions": true, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": "2"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Find the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": true, "correctAnswer": "inverse(b)", "markPerCell": false, "allowResize": false, "correctAnswerFractions": true, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": "2"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\mathbf{B}^{-1} = $&nbsp;[[0]]</p>", "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": true, "correctAnswer": "inverse(c)", "markPerCell": false, "allowResize": false, "correctAnswerFractions": true, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": "2"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\mathbf{C}^{-1} = $&nbsp;[[0]]</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Let</p>\n<p>\\begin{align} \\mathbf{A} &amp;= \\var{a}, &amp; \\mathbf{B} &amp;= \\var{b}, &amp; \\mathbf{C} &amp;= \\var{c} \\end{align}</p>", "tags": ["checked2015", "determinant of a matrix", "inverse", "inverse matrix", "MAS1602", "matrices", "matrix", "matrix inverse", "matrix multiplication", "multiplication of matrices", "tested1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>10/07/2012:</strong></p>\n        <p>Added tags.<strong><br /></strong></p>\n        <p>Question appears to be working correctly.</p>\n        <p>Corrected a typo in the Advice section.<strong><br /></strong></p>\n        <p><strong>24/12/2012:</strong></p>\n        <p>Checked calculations, OK. Added tested1 tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the determinant and inverse of three $2 \\times 2$ invertible matrices.</p>"}, "advice": "<h4>Determinant of a $2 \\times 2$ matrix</h4>\n<p>The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&amp;b \\\\ c&amp;d \\end{pmatrix}$ is given by</p>\n<p>\\[ \\det\\left(\\mathrm{M}\\right) = ad-bc \\]</p>\n<p>If we have two $n \\times n$ matrices $M$ and $N$, then</p>\n<p>\\[ \\det\\left(\\mathrm{MN}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right) \\]</p>\n<p>And it follows that if we have a third matrix $P$,</p>\n<p>\\[ \\det\\left(\\mathrm{MNP}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right)\\det\\left(\\mathrm{P}\\right) \\]</p>\n<h4>a)</h4>\n<p>Thus for our example we have:</p>\n<p>\\begin{align}<br/>\\det\\left(\\mathrm{A}\\right) &amp;= \\simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\\\<br/>\\det\\left(\\mathrm{B}\\right) &amp;= \\simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\\\<br/>\\det\\left(\\mathrm{C}\\right) &amp;= \\simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}}<br/>\\end{align}</p>\n<p>\\begin{align}<br/>\\det\\left( \\mathrm{ABC} \\right) &amp;= \\det(\\mathrm{A}) \\det(\\mathrm{B}) \\det(\\mathrm{C}) \\\\<br/>&amp;= \\simplify[]{{det(a)}*{det(b)}*{det(c)}} \\\\<br/>&amp;= \\var{det(a*b*c)}<br/>\\end{align}</p>\n<h4>Inverse of a $2 \\times 2$ matrix</h4>\n<p>Suppose $\\mathrm{M} = \\begin{pmatrix} a&amp;b \\\\ c&amp;d \\end{pmatrix}$ is a $2 \\times 2$ matrix and $\\det\\left(\\mathrm{M}\\right) = \\Delta \\neq 0$.</p>\n<p>Then $\\mathrm{M}$ is invertible and</p>\n<p>\\[ \\mathrm{M}^{-1} = \\frac{1}{\\Delta} \\begin{pmatrix} d &amp; -b\\\\ -c&amp; a \\end{pmatrix}=\\begin{pmatrix} \\frac{d}{\\Delta} &amp; -\\frac{b}{\\Delta}\\\\ -\\frac{c}{\\Delta}&amp; \\frac{a}{\\Delta} \\end{pmatrix}\\]</p>\n<p>Applying this to these examples we obtain:</p>\n<h4>b)</h4>\n<p>\\[ \\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \\]</p>\n<h4>c)</h4>\n<p>\\[ \\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \\]</p>\n<h4>d)</h4>\n<p>\\[ \\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \\]</p>"}, {"name": "Find eigenvalues and eigenvectors of matrices, and power of one matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "mxb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(a1,b1)", "description": "", "name": "mxb"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "name": "a"}, "x1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s*(b22-mnB)", "description": "", "name": "x1"}, "a22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+b-a11", "description": "", "name": "a22"}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s*(b22-mxB)", "description": "", "name": "x2"}, "cn22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mxB^n*x1-mnB^n*x2", "description": "", "name": "cn22"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "name": "a1"}, "cn11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mnB^n*x1-mxB^n*x2", "description": "", "name": "cn11"}, "da": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{a11*a22-a12*a21}", "description": "", "name": "da"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "name": "c2"}, "this": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mxA<0,mxA+random(1..6),mxA-random(1..6))", "description": "", "name": "this"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(4,5,6)", "description": "", "name": "n"}, "trb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{b11+b22}", "description": "", "name": "trb"}, "b11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(test1=a1,that,test1=b1,that,test1)", "description": "", "name": "b11"}, "tra": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{a11+a22}", "description": "", "name": "tra"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c1=a,a+random(1..3),c1)", "description": "", "name": "b"}, "cn12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "x1*x2*(mxB^n-mnB^n)", "description": "", "name": "cn12"}, "bn21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(s*cn21/f)", "description": "", "name": "bn21"}, "bn11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(s*cn11/f)", "description": "", "name": "bn11"}, "cn21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mnB^n-mxB^n", "description": "", "name": "cn21"}, "b21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-s", "description": "", "name": "b21"}, "mxa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(a,b)", "description": "", "name": "mxa"}, "b22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1-b11", "description": "", "name": "b22"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(x1-x2)", "description": "", "name": "f"}, "that": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mxB<0,mxB+random(1..6),mxB-random(1..6))", "description": "", "name": "that"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c2=a1,a1+random(1..3),c2)", "description": "", "name": "b1"}, "b12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s*(a1*b1-b11*a1-b11*b1+b11^2)", "description": "", "name": "b12"}, "mnb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "min(a1,b1)", "description": "", "name": "mnb"}, "test": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "name": "test"}, "db": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{b11*b22-b12*b21}", "description": "", "name": "db"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "name": "c1"}, "bn12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(s*cn12/f)", "description": "", "name": "bn12"}, "test1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "name": "test1"}, "mna": {"templateType": "anything", "group": "Ungrouped variables", "definition": "min(a,b)", "description": "", "name": "mna"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s"}, "a12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s", "description": "", "name": "a12"}, "a11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(test=a,this,test=b,this,test)", "description": "", "name": "a11"}, "a21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-s*(a*b-a11*a-a11*b+a11^2)", "description": "", "name": "a21"}, "bn22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(s*cn22/f)", "description": "", "name": "bn22"}}, "ungrouped_variables": ["test1", "a21", "a22", "bn22", "bn21", "b22", "b21", "b1", "cn11", "mna", "mnb", "cn12", "s2", "s1", "a12", "b12", "b11", "test", "cn21", "cn22", "a11", "that", "db", "bn12", "bn11", "da", "a1", "x2", "c2", "c1", "x1", "mxb", "mxa", "a", "b", "f", "tra", "trb", "n", "this", "s"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{mnA}", "minValue": "{mnA}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{mxA}", "minValue": "{mxA}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Find the eigenvalues of $A$.</p>\n<p>Let $a_1$ be the least eigenvalue of $A,\\;\\;\\; a_1=\\;\\;$[[0]]</p>\n<p>Let $a_2$ be the greatest eigenvalue of $A,\\;\\; a_2=\\;\\;$[[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{s*(mnA-a11)}", "minValue": "{s*(mnA-a11)}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{s*(mxA-a11)}", "minValue": "{s*(mxA-a11)}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Find eigenvectors for $A$.</p>\n      <p>Let $(1,y_1)^T$ be an eigenvector corresponding to $a_1,\\;\\;\\;\\;y_1=\\;\\;$[[0]]</p>\n      <p>Let $(1,y_2)^T$ be an eigenvector corresponding to $a_2,\\;\\;\\;\\;y_2=\\;\\;$[[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{mnB}", "minValue": "{mnB}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{mxB}", "minValue": "{mxB}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n        \n        \n        <p>Find the eigenvalues of $B$.</p>\n        \n        \n        \n        <p>Let $b_1$ be the least eigenvalue of $B,\\;\\;\\; b_1=\\;\\;$[[0]]</p>\n        \n        \n        \n        <p>Let $b_2$ be the greatest eigenvalue of $B,\\;\\; b_2=\\;\\;$[[1]]</p>\n        \n        \n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{x1}", "minValue": "{x1}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{x2}", "minValue": "{x2}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Find eigenvectors for $B$.</p>\n      <p>Let $(x_1,1)^T$ be an eigenvector corresponding to $b_1,\\;\\;\\;\\;x_1=\\;\\;$[[0]]</p>\n      <p>Let $(x_2,1)^T$ be an eigenvector corresponding to $b_2,\\;\\;\\;\\;x_2=\\;\\;$[[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{bn11}", "minValue": "{bn11}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{bn12}", "minValue": "{bn12}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{bn21}", "minValue": "{bn21}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{bn22}", "minValue": "{bn22}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <p>Find $B^{\\var{n}}$ using the last two parts of this question:</p>\n      <table>\n      <tbody>\n      <tr>\n      <td rowspan=\"2\">$B^{\\var{n}} = \\Bigg($</td>\n      <td>[[0]]</td>\n      <td>[[1]]</td>\n      <td rowspan=\"2\">$\\Bigg)$</td>\n      </tr>\n      <tr>\n      <td>[[2]]</td>\n      <td>[[3]]</td>\n      </tr>\n      </tbody>\n      </table>\n      <p>Input your answers as integers.</p>\n      ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n    \n    \n    <p>Find the eigenvalues and eigenvectors for the matrices $A$ and $B$  where:<br/>\\[ A=\\begin{pmatrix} \\var{a11}&#38;\\var{a12}\\\\ \\var{a21}&#38;\\var{a22} \\end{pmatrix},\\;\\;\\;\\;\\;\n    \n    B=\\begin{pmatrix} \\var{b11}&#38;\\var{b12}\\\\ \\var{b21}&#38;\\var{b22} \\end{pmatrix}\n    \n    \\]</p>\n    \n    \n  ", "tags": ["checked2015", "diagonalising matrices.", "eigenvalues", "eigenvalues of matrix", "eigenvectors of matrix", "MAS1602", "matrices", "matrix", "matrix eigenvalues", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>10/07/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>In the Advice section it is not explained how to find the trace and the determinant of the matrix - Should this be included?</p>\n        <p>Question appears to be working correctly.</p>\n        <p><strong>24/12/2012:</strong></p>\n        <p>Checked calculations, OK. Added tested1 tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$A,\\;B$ $2 \\times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n<h3>Matrix $A$</h3>\n<p>\\[A - \\lambda I_2 = \\begin{pmatrix} \\var{a11}-\\lambda &amp; \\var{a12}\\\\ \\var{a21} &amp; \\var{a22}-\\lambda \\end{pmatrix}\\]<br/>Hence the characteristic polynomial $p(\\lambda)$ is: \\[\\begin{eqnarray*} \\mathrm{det}\\left(A-\\lambda I_2 \\right)&amp;=&amp;\\simplify[zeroTerm]{({a11}-lambda)({a22}-lambda)-{a12}*{a21}}\\\\ &amp;=&amp; \\simplify[std]{lambda^2-{trA}*lambda+{dA}}\\\\ &amp;=&amp;\\simplify[std]{(lambda-{a})(lambda-{b})} \\end{eqnarray*} \\]<br/>We see that on solving $p(\\lambda)=0$ we get the eigenvalues:<br/>\\[\\lambda_1=\\var{mnA},\\;\\;\\;\\lambda_2=\\var{mxA}\\] <br/>Note: We could have found the characteristic polynomial by noting that for a 2 &times; 2 matrix $A$ then the characteristic polynomial is<br/>\\[\\lambda^2-\\mathrm{trace}(A)+\\mathrm{det}(A)\\]<br/>where $\\mathrm{trace}(A) = \\var{trA},\\;\\;\\;\\mathrm{det}(A)=\\var{dA}$</p>\n<p>b)</p>\n<h4>Finding the eigenvectors:</h4>\n<p>1. $\\lambda=\\var{mnA}$</p>\n<p>We have the eigenspace is given by all $v=(x,y)^T$ such that $(\\simplify{A-{mnA}}I_2)v=(0,0)^T$ i.e.</p>\n<p>\\[\\begin{pmatrix} \\var{a11-mnA}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22-mnA} \\end{pmatrix}\\begin{pmatrix} x \\\\ y \\end{pmatrix} =\\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}\\]</p>\n<p>This gives the two equations:</p>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{a11-mnA}x + {a12}y}&amp;=&amp;0\\\\ \\simplify[std]{{a21}x + {a22-mnA}y}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>There is only one equation here as we see that the equations are the same (one is a multiple of the other).</p>\n<p>So putting $x=1$ in the first equation we get $y_1=\\var{-s*(a11-mnA)}$</p>\n<p>Hence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mnA)} \\end{pmatrix}\\]</p>\n<p>2. $\\lambda=\\var{mxA}$</p>\n<p>In this case we have the equations:</p>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{a11-mxA}x + {a12}y}&amp;=&amp;0\\\\ \\simplify[std]{{a21}x + {a22-mxA}y}&amp;=&amp;0 \\end{eqnarray*} \\]</p>\n<p>Once again there is only one equation, so putting $x=1$ in the first equation we get $y_2=\\var{-s*(a11-mxA)}$</p>\n<p>Hence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mxA)} \\end{pmatrix}\\]</p>\n<p>c)</p>\n<h3>Matrix $B$</h3>\n<p>The characteristic polynomial is given by:</p>\n<p>\\[p(\\lambda)=\\simplify[std]{lambda^2-{b11+b22}*lambda + {dB}}\\]</p>\n<p>Solving $p(\\lambda)=0$, we find the eigenvalues for $B$ are:<br/>\\[\\lambda_1=\\var{mnB},\\;\\;\\;\\lambda_2=\\var{mxB}\\]</p>\n<p>d)</p>\n<h4>Eigenvectors</h4>\n<p>1. $\\lambda=\\var{mnB}$</p>\n<p>The equations are:<br/>\\[ \\begin{eqnarray*} \\simplify[std]{{b11-mnB}x + {b12}y}&amp;=&amp;0\\\\ \\simplify[std]{{b21}x + {b22-mnB}y}&amp;=&amp;0 \\end{eqnarray*} \\]</p>\n<p>Putting $y=1$ in the second equation we get $x_1=\\var{s*(b22-mnB)}$</p>\n<p>Hence the eigenvector we want is \\[\\begin{pmatrix} \\var{s*(b22-mnB)}\\\\1 \\end{pmatrix}\\]</p>\n<p>2. $\\lambda=\\var{mxB}$</p>\n<p>The equations are:<br/>\\[ \\begin{eqnarray*} \\simplify[std]{{b11-mxB}x + {b12}y}&amp;=&amp;0\\\\ \\simplify[std]{{b21}x + {b22-mxB}y}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>Putting $y=1$ in the second equation we get $x_2=\\var{s*(b22-mxB)}$</p>\n<p>Hence the eigenvector we want is \\[\\begin{pmatrix} \\var{s*(b22-mxB)}\\\\1 \\end{pmatrix}\\]</p>\n<p>e)</p>\n<p>For the last part we use the diagonalisation of $B$ given by the last two parts.</p>\n<p>Thus if $x_1,\\;\\;x_2,\\;\\;\\lambda_1,\\;\\;\\lambda_2$ are as above for $B$ then we have $B=PDP^{-1} \\Rightarrow B^{\\var{n}}=PD^{\\var{n}}P^{-1}$ where:</p>\n<p>\\[\\begin{eqnarray*} P &amp;=&amp; \\begin{pmatrix} x_1 &amp; x_2\\\\1&amp;1 \\end{pmatrix} = \\begin{pmatrix} \\var{s*(b22-mnB)} &amp; \\var{s*(b22-mxB)} \\\\1&amp;1 \\end{pmatrix}\\Rightarrow P^{-1}= \\simplify[std]{1/{x1-x2}}\\begin{pmatrix} 1 &amp; \\var{-s*(b22-mxB)} \\\\-1&amp;\\var{s*(b22-mnB)} \\end{pmatrix}\\\\ \\\\ D&amp;=&amp; \\begin{pmatrix} \\lambda_1 &amp; 0\\\\0&amp;\\lambda_2 \\end{pmatrix} = \\begin{pmatrix} \\var{mnB} &amp; 0\\\\0&amp;\\var{mxB} \\end{pmatrix} \\Rightarrow D^{\\var{n}}=\\begin{pmatrix} \\var{mnB^n} &amp; 0\\\\0&amp;\\var{mxB^n} \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<p>Hence \\[\\begin{eqnarray*}B^{\\var{n}}&amp;=&amp;PD^{\\var{n}}P^{-1}\\\\ \\\\ &amp;=&amp;\\simplify[std]{1/{x1-x2}}\\begin{pmatrix} \\var{s*(b22-mnB)} &amp; \\var{s*(b22-mxB)} \\\\1&amp;1 \\end{pmatrix}\\begin{pmatrix} \\var{mnB^n} &amp; 0\\\\0&amp;\\var{mxB^n} \\end{pmatrix}\\begin{pmatrix} 1 &amp; \\var{-s*(b22-mxB)} \\\\-1&amp;\\var{s*(b22-mnB)} \\end{pmatrix}\\\\ \\\\ &amp;=&amp;\\begin{pmatrix} \\var{bn11} &amp; \\var{bn12}\\\\\\var{bn21}&amp;\\var{bn22} \\end{pmatrix} \\end{eqnarray*} \\]</p>"}, {"name": "Find eigenvalues, characteristic polynomial and a normalised eigenvector of a 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"eigen3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except[0,eigen1,eigen2])", "description": "", "name": "eigen3"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1-g*a", "description": "", "name": "b"}, "invp": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([0,1,g],[a,0,1],[b,1,0])", "description": "", "name": "invp"}, "det": {"templateType": "anything", "group": "Ungrouped variables", "definition": "eigen1*eigen2*eigen3", "description": "", "name": "det"}, "coefflam": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-(eigen1*eigen2+eigen1*eigen3+eigen2*eigen3)", "description": "", "name": "coefflam"}, "M": {"templateType": "anything", "group": "Ungrouped variables", "definition": "p*D*invp", "description": "", "name": "M"}, "eigen2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except [0,eigen1])", "description": "", "name": "eigen2"}, "coefflamsq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "eigen1+eigen2+eigen3", "description": "", "name": "coefflamsq"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2..2 except 0)", "description": "", "name": "a"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "g"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([-1,g,1],[b,-g*b,1-b],[a,b,-a])", "description": "", "name": "p"}, "eigen1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "eigen1"}, "D": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([eigen1,0,0],[0,eigen2,0],[0,0,eigen3])", "description": "", "name": "D"}}, "ungrouped_variables": ["a", "b", "D", "g", "det", "M", "coefflamsq", "invp", "p", "coefflam", "eigen2", "eigen3", "eigen1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "eigen1", "minValue": "eigen1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "eigen3", "minValue": "eigen3", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>You are given that $\\var{vector(1,-b,-a)}$ is an eigenvector of $M$.</p>\n<p>Find the corresponding eigenvalue: [[0]]</p>\n<p>Also $\\var{vector(1,1-b,-a)}$ is an eigenvector of $M$.</p>\n<p>Find the corresponding eigenvalue: [[1]]</p>\n<p></p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "-lambda^3+{coefflamsq}*lambda^2+{coefflam}*lambda+{det}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "eigen2", "minValue": "eigen2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{-b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{b}/{g}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Enter numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all,fractionNumbers", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Find the characteristic polynomial of $M$ and hence another eigenvalue for $M$.</p>\n<p>Enter the characteristic polynomial in the form $P_M(\\lambda) = -\\lambda^3+a\\lambda^2+b\\lambda+c$.</p>\n<p>Write the letter $\\lambda$ as <code>lambda</code>.</p>\n<p>Characteristic polynomial: $P_M(\\lambda) = \\;$[[0]]</p>\n<p>Hence find another eigenvalue of $M$: [[1]]</p>\n<p>Find a corresponding eigenvector for this eigenvalue. Scale your vector such that the first component is $1$. You have to find the other two components:</p>\n<p>Eigenvector = $\\Bigg($ <span class=\"vector-input\"><span class=\"component\">$1$</span><span class=\"component\">[[2]]</span><span class=\"component\">[[3]]</span></span> $\\Bigg)$</p>\n<p>Input both components as fractions or integers and not as decimals.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let $M=\\var{M}$. &nbsp;Answer the following questions.</p>", "tags": ["characteristic equation", "characteristic polynomial", "checked2015", "determinant", "eigenvalues", "eigenvectors", "linear algebra", "linear equations", "MAS1602", "matrices", "matrix algebra"], "rulesets": {}, "preamble": {"css": ".vector-input {\n  display: inline-block;\n  vertical-align: middle;\n  text-align: center;\n}\n.vector-input .component {\n  float: center;\n  clear: both;\n  display: block !important;\n  text-align:  center;\n}\n.vector-input .part {\n  margin-top: 0;\n  margin-bottom: 0;\n}", "js": ""}, "type": "question", "metadata": {"notes": "<p>Created 19/09/2014</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h3>a)</h3>\n<p>For the eigenvector $\\var{vector(1,-b,-a)}$ we have:</p>\n<p>$\\var{M}\\var{vector(1,-b,-a)}=\\var{vector(eigen1,-eigen1*b,-eigen1*a)}=\\var{eigen1}\\var{vector(1,-b,-a)}$.</p>\n<p>Hence the corresponding eigenvalue is $\\var{eigen1}$.</p>\n<p>Similarly, for the eigenvector $\\var{vector(1,1-b,-a)}$ we have:</p>\n<p>$\\var{M}\\var{vector(1,1-b,-a)}=\\var{vector(eigen3,eigen3*(1-b),-eigen3*a)}=\\var{eigen3}\\var{vector(1,1-b,-a)}$.</p>\n<p>Hence the corresponding eigenvalue for this eigenvector is $\\var{eigen3}$.</p>\n<h3>b)</h3>\n<p>The characteristic polynomial is given by $P_M(\\lambda)=\\operatorname{det}(M-\\lambda I_3)$. The roots of this are the eigenvalues.</p>\n<p>We find that in this case:</p>\n<p>\\[\\begin{align}\\operatorname{det}(M-\\lambda I_3)&amp;=\\operatorname{det}\\begin{pmatrix}\\var{m[0][0]}-\\lambda &amp;\\var{m[0][1] }&amp;\\var{m[0][2]}\\\\ \\var{m[1][0]}&amp;\\var{m[1][1] }-\\lambda&nbsp;&amp;\\var{m[1][2]}\\\\ &nbsp;\\var{m[2][0]}&amp;\\var{m[2][1] } &amp;\\var{m[2][2]}-\\lambda \\end{pmatrix}\\\\&amp;=\\simplify{-lambda^3+{coefflamsq}*lambda^2+{coefflam}*lambda+{det}}\\end{align}\\]</p>\n<p>Now we know that $\\simplify{lambda-{eigen1}}$ and &nbsp;$\\simplify{lambda-{eigen3}}$ are both factors of the characteristic polynomial.</p>\n<p>Hence we have:</p>\n<p>$\\simplify{-lambda^3+{coefflamsq}*lambda^2+{coefflam}*lambda+{det}=-(lambda-{eigen1})(lambda-{eigen3})(lambda-r)}$ for some number $r$.</p>\n<p>Since the constant term in the characteristic polynomial is the product of the three eigenvalues, $r=\\simplify[!basic]{{det}/({eigen1}*{eigen3})={eigen2}}$ and this is the third eigenvalue.</p>\n<p>To find an eigenvector corresponding to this eigenvalue we solve the equation:</p>\n<p>$\\simplify{M*vector(x,y,z)={M}*vector(x,y,z)={eigen2}*vector(x,y,z)}$</p>\n<p>This gives the equations:</p>\n<p>\\[\\begin{align} \\simplify{{m[0][0]}*x+{m[0][1]}*y+{m[0][2]}*z}&amp;=\\simplify{{eigen2}*x}\\\\ \\simplify{{m[1][0]}*x+{m[1][1]}*y+{m[1][2]}*z}&amp;=\\simplify{{eigen2}*y}\\\\ \\simplify{{m[2][0]}*x+{m[2][1]}*y+{m[2][2]}*z}&amp;=\\simplify{{eigen2}*z}\\end{align}\\]</p>\n<p>The question asked you to find an eigenvector with $x=1$ and if we substitute this into the equations we find (only need to use two of the equations) that $y=\\var{-b}$ and $z=\\simplify[all,fractionNumbers]{{b}/{g}}$.</p>\n<p>Hence the eigenvector we want is $\\simplify[all,fractionNumbers]{vector(1,{-b},{b}/{g})}$.</p>"}, {"name": "Find the Determinant of a 3 x 3 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["a"], "name": "Unnamed group"}], "variables": {"a": {"templateType": "anything", "group": "Unnamed group", "definition": "matrix([ [a11,a12,a13],[a21,a22,a23],[a31,a32,a33] ])", "description": "", "name": "a"}, "a13": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "name": "a13"}, "a22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "<p>Matrix element</p>", "name": "a22"}, "a31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "name": "a31"}, "a23": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "name": "a23"}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a21*a32)-(a22*a31)", "description": "<p>Submatrix</p>", "name": "m3"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a22*a33)-(a23*a32)", "description": "<p>Submatrix</p>", "name": "m1"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a21*a33)-(a23*a31)", "description": "<p>Submatrix</p>", "name": "m2"}, "a32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "name": "a32"}, "a12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "name": "a12"}, "a11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "<p>Matrix element</p>", "name": "a11"}, "a33": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "<p>Matrix element</p>", "name": "a33"}, "a21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "<p>Matrix element</p>", "name": "a21"}}, "ungrouped_variables": ["a11", "a12", "a13", "a21", "a22", "a23", "a31", "a32", "a33", "m1", "m2", "m3"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>Calculate the determinant of the matrix.</p>\n<p>$\\operatorname{det}\\left( \\mathrm{A}\\right) = $&nbsp;[[0]]</p>\n<p></p>", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "det(a)", "minValue": "det(a)", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>Consider the&nbsp;$3 \\times 3$ matrix,</p>\n<p>\\begin{align} \\mathrm{A} &amp;= \\var{a} \\end{align}</p>", "tags": ["determinant of a matrix", "MAS1602", "mas1602", "matrices", "matrix"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the determinant of a&nbsp;$3 \\times 3$ matrix.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>Determinant of a $3 \\times 3$ matrix</h4>\n<p>The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&amp;b&amp;c \\\\ d&amp;e&amp;f \\\\ g&amp;h&amp;i \\end{pmatrix}$&nbsp;can be calculated&nbsp;by using cofactor expansion. Expanding along the first row,</p>\n<p>\\[ \\det\\left(\\mathrm{M}\\right) = a \\cdot \\det&nbsp;\\begin{pmatrix} e&amp;f \\\\ h&amp;i \\end{pmatrix}- b&nbsp;\\cdot \\det&nbsp;\\begin{pmatrix} d&amp;f \\\\ g&amp;i \\end{pmatrix} + c&nbsp;\\cdot \\det&nbsp;\\begin{pmatrix} d&amp;e \\\\ g&amp;h \\end{pmatrix}\\]</p>\n<p>Thus for our example we have:</p>\n<p>\\[\\begin{align} \\det&nbsp;\\begin{pmatrix} e&amp;f \\\\ h&amp;i \\end{pmatrix} &amp;= \\simplify[]{({a22}*{a33})-({a23}*{a32})&nbsp;= {m1}}&nbsp;\\\\ \\det&nbsp;\\begin{pmatrix} d&amp;f \\\\ g&amp;i \\end{pmatrix} &amp;= \\simplify[]{({a21}*{a33})-({a23}*{a31})&nbsp;=&nbsp;{m2}} \\\\ \\det&nbsp;\\begin{pmatrix} d&amp;e \\\\ g&amp;h \\end{pmatrix} &amp;=\\simplify[]{ ({a21}*{a32})-({a22}*{a31})&nbsp;={m3}} &nbsp;\\end{align}\\]</p>\n<p>and so</p>\n<p>\\[\\begin{align} \\det\\left(\\mathrm{A}\\right) = (\\simplify[]{{a11}*{m1}})-(\\simplify[]{{a12}*{m2}})+(\\simplify[]{{a13}*{m3}}) = \\simplify[]{{det(a)}} &nbsp;\\end{align}\\]</p>"}, {"name": "Find the Determinant of a 4 x 4 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["determinant of a matrix", "matrices", "matrix"], "metadata": {"description": "<p>Find the determinant of a&nbsp;$4 \\times 4$ matrix.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the&nbsp;$4 \\times 4$ matrix,</p>\n<p>\\begin{align} \\mathrm{A} &amp;= \\var{a} \\end{align}</p>", "advice": "<p>The determinant of an $n \\times n$ matrix an be calculated in terms of the determinant of minor matrices of size $[n-1] \\times [n-1]$.&nbsp;</p>\n<p>For our example we multiply&nbsp;each value in a chosen row of the matrix by its cofactor,</p>\n<p>\\[&nbsp;(-1)^{i+j} M_{ij} \\]</p>\n<p>for a&nbsp;value at row $i$ and column $j$, where $M_{ij}$ is its&nbsp;minor $3 \\times 3$ matrix. Adding these values together gives the determinant of our $4 \\times 4$ matrix.</p>\n<p>In our example, choosing the first row we have,</p>\n<p>\\[&nbsp;\\det\\left(\\mathrm{A}\\right) = \\var{a11} \\cdot \\det \\var{m1}-\\var{a12} \\cdot \\det \\var{m2}+\\var{a13} \\cdot \\det \\var{m3} - \\var{a14} \\cdot \\det \\var{m4} \\]</p>\n<p>Notice that we do not need to compute all of the 3 x 3 matrices, particularly if the starting row is chosen carefully.</p>\n<p>Then,</p>\n<p>\\[&nbsp;\\det\\left(\\mathrm{A}\\right) = (\\simplify[]{{a11}*{det(m1)}})+(\\simplify[]{{a14}*{det(m4)}}) = \\var{determinant} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a41": {"name": "a41", "group": "Matrix Elements", "definition": "0", "description": "", "templateType": "anything", "can_override": false}, "m4": {"name": "m4", "group": "Minor Matrices", "definition": "matrix([ 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"m4"]}, {"name": "Matrix Elements", "variables": ["a11", "a12", "a13", "a14", "a21", "a22", "a23", "a24", "a31", "a32", "a33", "a34", "a41", "a42", "a43", "a44"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the determinant of the matrix through expansion by&nbsp;the first row.</p>\n<p>$\\det&nbsp;\\left(\\mathrm{A} \\right) = $&nbsp;[[0]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.5, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], 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"prompt": "\n      <h4>Part 1.</h4>\n        <p>Introduce zeros in the first column below the first entry by adding suitable multiples of the first row to rows 2 and 3.</p>\n        <p>Input all numbers as fractions or integers and not as decimals.</p>\n        <p>&nbsp;</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$\\var{a11}$</td>\n        <td>$\\var{a12}$</td>\n        <td>$\\var{a13}$</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>[[0]]</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        <td>[[5]]</td>\n        <td>$0$</td>\n        <td>$1$</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>Now, if necessary, multiply the second row by a suitable number so that the second entry in the second row is 1.</p>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{s*(-b*c+a*b*c)}", "minValue": "{s*(-b*c+a*b*c)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a*g2*g3}", "minValue": "{a*g2*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ g2*a*f1}", "minValue": "{ g2*a*f1}", "correctAnswerFraction": 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<h4>Part 2.</h4>\n        <p>Now using this matrix, introduce a zero in the second column below the second entry of the second column by:</p>\n        <p>Adding [[0]] times the second row to the third row to get the matrix:</p>\n        <p>&nbsp;</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$\\var{a11}$</td>\n        <td>$\\var{a12}$</td>\n        <td>$\\var{a13}$</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$1$</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        <td>$0$</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[4]]</td>\n        <td>[[5]]</td>\n        <td>[[6]]</td>\n        <td>$1$</td>\n        </tr>\n        </tbody>\n        </table>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a*g2*g3}", "minValue": "{-a*g2*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a13}", "minValue": "{-a13}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ 1-a13*(a*b-b-1)*g3*a*c*f1-a12*b24}", "minValue": "{ 1-a13*(a*b-b-1)*g3*a*c*f1-a12*b24}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a13*f2*g3*b*c*(a-1)-a12*b25}", "minValue": "{a13*f2*g3*b*c*(a-1)-a12*b25}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a13*f3*g3+a12*a*g2*f3}", "minValue": "{-a13*f3*g3+a12*a*g2*f3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{b24}", "minValue": "{b24}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{b25}", "minValue": "{b25}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a*f3*g2}", "minValue": "{-a*f3*g2}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{(a*b-b-1)*g3*a*c*f1}", "minValue": "{(a*b-b-1)*g3*a*c*f1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ -f2*g3*b*c*(a-1)}", "minValue": "{ -f2*g3*b*c*(a-1)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{f3*g3}", "minValue": "{f3*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <h4>Part 3.</h4>\n        <p>Now, if necessary, multiply the third row by a suitable constant so that the third entry in the third column is 1.</p>\n        <p>With this matrix, use the third row to introduce zeros into the second and first entries in the third column by adding suitable multiples of the third row to the second and first rows.</p>\n        <p>Multiply third row by [[0]]and add to the second row.</p>\n        <p>Multiply third row by [[1]]and add to the first row.</p>\n        <p>Using this new matrix there is one more operation needed.</p>\n        <p>Multiplying the second row by $\\var{-a12}$ and adding to the first row to obtain the inverse matrix appearing on the right hand side.</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>[[5]]</td>\n        <td>[[6]]</td>\n        <td>[[7]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$1$</td>\n        <td>[[8]]</td>\n        <td>[[9]]</td>\n        <td>[[10]]</td>\n        </tr>\n        </tbody>\n        </table>\n      ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n  <p>Find the inverse of the following matrix:<br/>\\[A = \\left(\\begin{array}{rrr} \\var{a11} &amp; \\var{a12} &amp; \\var{a13}\\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23}\\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33}\\\\ \\end{array}\\right)\\]</p>\n    <p>Form the $3 \\times 6$ augmented matrix $B$ by placing $I_3$ to the right of $A$ as below:<br/>\\[B = \\left(\\begin{array}{rrr|ccc} \\var{a11} &amp; \\var{a12} &amp; \\var{a13} &amp;\\var{1}&amp;\\var{0}&amp;\\var{0}\\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23}&amp;\\var{0}&amp;\\var{1}&amp;\\var{0}\\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33}&amp;\\var{0}&amp;\\var{0}&amp;\\var{1}\\\\ \\end{array}\\right)\\]</p>\n    <p><strong>In subsequent parts work with this matrix using row operations and introduce the identity matrix on the left hand side with the inverse of A eventually appearing on the right hand side.</strong></p>\n  ", "tags": ["MAS1602", "augmented matrices", "augmented matrix", "checked2015", "inverse matrix", "invertible matrices", "inverting a matrix", "linear algebra", "matrices", "matrix inverse", "matrix inversion"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n            \t\t<p><strong>5/07/2012:</strong></p>\n            \t\t<p>Added tags.</p>\n            \t\t<p>Changed grammar in the question.</p>\n            \t\t<p>Question appears to be working correctly.</p>\n            \t\t<p><strong>14/07/2012:</strong></p>\n            \t\t<p>Need to align columns where input takes place through the stages.</p>\n            \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$A$ a $3 \\times 3$ matrix. Using row operations on the augmented matrix $\\left(A | I_3\\right)$ reduce to $\\left(I_3 | A^{-1}\\right)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>All of the working is now shown</p>"}, {"name": "Invert a 3x3 matrix using row operations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "a"}, "f1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "f1"}, "a22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f2*(a*b-1)*g2", "description": "", "name": "a22"}, "g3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "g3"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "c2"}, "f2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "f2"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "c3"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "b"}, "f3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "f3"}, "a33": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f3*g3", "description": "", "name": "a33"}, "b24": {"templateType": "anything", "group": "Ungrouped variables", "definition": "g2*a*f1*(1-(a*b-b-1)*a*c)", "description": "", "name": "b24"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,3)", "description": "", "name": "c"}, "g1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f1", "description": "", "name": "g1"}, "g2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "g2"}, "b25": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f2*g2*(a*(a*b*c-b*c)-1)", "description": "", "name": "b25"}, "a31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f3*a*c*g1", "description": "", "name": "a31"}, "a23": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f2*(a^2*b-a-a*b)*g3", "description": "", "name": "a23"}, "a13": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f1*(b*a-b)*g3", "description": "", "name": "a13"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "c1"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f3*g2", "description": "", "name": "s"}, "a32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f3*c*b*g2", "description": "", "name": "a32"}, "a12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f1*b*g2", "description": "", "name": "a12"}, "a11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f1*g1", "description": "", "name": "a11"}, "a21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f2*a*g1", "description": "", "name": "a21"}}, "ungrouped_variables": ["a", "a11", "a12", "a13", "a21", "a22", "a23", "a31", "a32", "a33", "b", "b24", "b25", "c", "c1", "c2", "c3", "f1", "f2", "f3", "g1", "g2", "g3", "s"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-f2*g2}", "minValue": "{-f2*g2}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a*f2*g3}", "minValue": "{-a*f2*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ -f2*f1*a}", "minValue": "{ -f2*f1*a}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{f3*g2*c*b*(1-a)}", "minValue": "{f3*g2*c*b*(1-a)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{f3*g3*(1-a^2*b*c+a*b*c)}", "minValue": "{f3*g3*(1-a^2*b*c+a*b*c)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-f3*f1*a*c}", "minValue": "{-f3*f1*a*c}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <h4>Part 1.</h4>\n        <p>Introduce zeros in the first column below the first entry by adding suitable multiples of the first row to rows 2 and 3.</p>\n        <p>Input all numbers as fractions or integers and not as decimals.</p>\n        <p>&nbsp;</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$\\var{a11}$</td>\n        <td>$\\var{a12}$</td>\n        <td>$\\var{a13}$</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>[[0]]</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        <td>[[5]]</td>\n        <td>$0$</td>\n        <td>$1$</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>Now, if necessary, multiply the second row by a suitable number so that the second entry in the second row is 1.</p>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{s*(-b*c+a*b*c)}", "minValue": "{s*(-b*c+a*b*c)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a*g2*g3}", "minValue": "{a*g2*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ g2*a*f1}", "minValue": "{ g2*a*f1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-f2*g2}", "minValue": "{-f2*g2}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{f3*g3}", "minValue": "{f3*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{(a*b-b-1)*f3*a*c*f1}", "minValue": "{(a*b-b-1)*f3*a*c*f1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-f2*f3*(a*b*c-b*c)}", "minValue": "{-f2*f3*(a*b*c-b*c)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <h4>Part 2.</h4>\n        <p>Now using this matrix, introduce a zero in the second column below the second entry of the second column by:</p>\n        <p>Adding [[0]] times the second row to the third row to get the matrix:</p>\n        <p>&nbsp;</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$\\var{a11}$</td>\n        <td>$\\var{a12}$</td>\n        <td>$\\var{a13}$</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$1$</td>\n        <td>[[1]]</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        <td>$0$</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[4]]</td>\n        <td>[[5]]</td>\n        <td>[[6]]</td>\n        <td>$1$</td>\n        </tr>\n        </tbody>\n        </table>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a*g2*g3}", "minValue": "{-a*g2*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a13}", "minValue": "{-a13}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ 1-a13*(a*b-b-1)*g3*a*c*f1-a12*b24}", "minValue": "{ 1-a13*(a*b-b-1)*g3*a*c*f1-a12*b24}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a13*f2*g3*b*c*(a-1)-a12*b25}", "minValue": "{a13*f2*g3*b*c*(a-1)-a12*b25}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a13*f3*g3+a12*a*g2*f3}", "minValue": "{-a13*f3*g3+a12*a*g2*f3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{b24}", "minValue": "{b24}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{b25}", "minValue": "{b25}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{-a*f3*g2}", "minValue": "{-a*f3*g2}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{(a*b-b-1)*g3*a*c*f1}", "minValue": "{(a*b-b-1)*g3*a*c*f1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{ -f2*g3*b*c*(a-1)}", "minValue": "{ -f2*g3*b*c*(a-1)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{f3*g3}", "minValue": "{f3*g3}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <h4>Part 3.</h4>\n        <p>Now, if necessary, multiply the third row by a suitable constant so that the third entry in the third column is 1.</p>\n        <p>With this matrix, use the third row to introduce zeros into the second and first entries in the third column by adding suitable multiples of the third row to the second and first rows.</p>\n        <p>Multiply third row by [[0]]and add to the second row.</p>\n        <p>Multiply third row by [[1]]and add to the first row.</p>\n        <p>Using this new matrix there is one more operation needed.</p>\n        <p>Multiplying the second row by $\\var{-a12}$ and adding to the first row to obtain the inverse matrix appearing on the right hand side.</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>[[2]]</td>\n        <td>[[3]]</td>\n        <td>[[4]]</td>\n        <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$1$</td>\n        <td>$0$</td>\n        <td>[[5]]</td>\n        <td>[[6]]</td>\n        <td>[[7]]</td>\n        </tr>\n        <tr>\n        <td>$0$</td>\n        <td>$0$</td>\n        <td>$1$</td>\n        <td>[[8]]</td>\n        <td>[[9]]</td>\n        <td>[[10]]</td>\n        </tr>\n        </tbody>\n        </table>\n      ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n  <p>Find the inverse of the following matrix:<br/>\\[A = \\left(\\begin{array}{rrr} \\var{a11} &amp; \\var{a12} &amp; \\var{a13}\\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23}\\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33}\\\\ \\end{array}\\right)\\]</p>\n    <p>Form the $3 \\times 6$ augmented matrix $B$ by placing $I_3$ to the right of $A$ as below:<br/>\\[B = \\left(\\begin{array}{rrr|ccc} \\var{a11} &amp; \\var{a12} &amp; \\var{a13} &amp;\\var{1}&amp;\\var{0}&amp;\\var{0}\\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23}&amp;\\var{0}&amp;\\var{1}&amp;\\var{0}\\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33}&amp;\\var{0}&amp;\\var{0}&amp;\\var{1}\\\\ \\end{array}\\right)\\]</p>\n    <p><strong>In subsequent parts work with this matrix using row operations and introduce the identity matrix on the left hand side with the inverse of A eventually appearing on the right hand side.</strong></p>\n  ", "tags": ["MAS1602", "augmented matrices", "augmented matrix", "checked2015", "inverse matrix", "invertible matrices", "inverting a matrix", "linear algebra", "matrices", "matrix inverse", "matrix inversion"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n            \t\t<p><strong>5/07/2012:</strong></p>\n            \t\t<p>Added tags.</p>\n            \t\t<p>Changed grammar in the question.</p>\n            \t\t<p>Question appears to be working correctly.</p>\n            \t\t<p><strong>14/07/2012:</strong></p>\n            \t\t<p>Need to align columns where input takes place through the stages.</p>\n            \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$A$ a $3 \\times 3$ matrix. Using row operations on the augmented matrix $\\left(A | I_3\\right)$ reduce to $\\left(I_3 | A^{-1}\\right)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>All of the working is now shown</p>"}, {"name": "Matrices: Manipulation", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "George Stagg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/930/"}], "tags": ["checked2015", "Linear algebra", "Linear Algebra", "linear algebra", "matrices", "matrix", "matrix manipulation", "matrix multiplication", "multiply matrix", "products of matrices"], "metadata": {"description": "<p>Elementary Exercises in multiplying matrices.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Answer the following questions on matrices.</p>\n    <p>&nbsp;</p>", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"u": {"name": "u", "group": "Ungrouped variables", "definition": "random(3..4)", "description": "", "templateType": "anything"}, "a23": {"name": "a23", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "", "templateType": "anything"}, "s5": {"name": "s5", "group": "Ungrouped variables", "definition": "if(m=3,0.5,-0.5)", "description": "", "templateType": "anything"}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "a11*v1+a12*v2+a13*v3", "description": "", "templateType": "anything"}, "b11": {"name": "b11", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(-4,-3,-2,-1,1,2,3,5)", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "if(r=2,0.5,-0.5)", "description": "", "templateType": "anything"}, "s7": {"name": "s7", "group": "Ungrouped variables", "definition": "if(u=y,0.5,-0.5)", "description": "", "templateType": "anything"}, "s8": {"name": "s8", "group": "Ungrouped variables", "definition": "if(x=r,0.5,-0.5)", "description": "", "templateType": "anything"}, "mad": {"name": "mad", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-2..9),q),p))", "description": "", "templateType": "anything"}, "b23": {"name": "b23", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "templateType": "anything"}, "mac": {"name": "mac", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-2..9),n),m))", "description": "", "templateType": "anything"}, "a33": {"name": "a33", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "mag": {"name": "mag", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-2..9),u),w))", "description": "", "templateType": "anything"}, "ba32": {"name": "ba32", "group": "Ungrouped variables", "definition": "b31*a12+b32*a22+b33*a32", "description": "", "templateType": "anything"}, "b21": {"name": "b21", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "n+random(0,z)", "description": "", "templateType": "anything"}, "ab33": {"name": "ab33", "group": "Ungrouped variables", "definition": "a31*b13+a32*b23+a33*b33", "description": "", "templateType": "anything"}, "mah": {"name": "mah", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-2..9),x),y))", "description": "", "templateType": "anything"}, "mae": {"name": "mae", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-2..9),s),r))", "description": "", "templateType": "anything"}, "ba22": {"name": "ba22", "group": "Ungrouped variables", "definition": "b21*a12+b22*a22+b23*a32", "description": "", "templateType": "anything"}, "a13": {"name": "a13", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "ab13": {"name": "ab13", "group": "Ungrouped variables", "definition": "a11*b13+a12*b23+a13*b33", "description": "", "templateType": "anything"}, "w2": {"name": "w2", "group": "Ungrouped variables", "definition": "random(-4..4)", "description": "", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "b11*w1+b12*w2+b13*w3", "description": "", "templateType": "anything"}, "a32": {"name": "a32", "group": "Ungrouped variables", "definition": "random(-4..4)", "description": "", "templateType": "anything"}, "ab23": {"name": "ab23", "group": "Ungrouped variables", "definition": "a21*b13+a22*b23+a23*b33", "description": "", "templateType": "anything"}, "ba11": {"name": "ba11", "group": "Ungrouped variables", "definition": "b11*a11+b12*a21+b13*a31", "description": "", "templateType": "anything"}, "ab32": {"name": "ab32", "group": "Ungrouped variables", "definition": "a31*b12+a32*b22+a33*b32", "description": "", "templateType": "anything"}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "ba31": {"name": "ba31", "group": "Ungrouped variables", "definition": "b31*a11+b32*a21+b33*a31", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(-1,0,1)", "description": "", "templateType": "anything"}, "v3": {"name": "v3", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "if(n=p,0.5,-0.5)", "description": "", "templateType": "anything"}, "ba33": {"name": "ba33", "group": "Ungrouped variables", "definition": "b31*a13+b32*a23+b33*a33", "description": "", "templateType": "anything"}, "v1": {"name": "v1", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "[[s1,-s1],[s2,-s2],[s3,-s3],[s4,-s4],[s5,-s5],[s6,-s6],[s7,-s7],[s8,-s8],[s9,-s9],[s10,-s10]]", "description": "", "templateType": "anything"}, "w1": {"name": "w1", "group": "Ungrouped variables", "definition": "random(4..6)", "description": "", "templateType": "anything"}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "random(-2,-1,1,2)", "description": "", "templateType": "anything"}, "c13": {"name": "c13", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "a31*v1+a32*v2+a33*v3", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "if(q=m,0.5,-0.5)", "description": "", "templateType": "anything"}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "b21*w1+b22*w2+b23*w3", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..4)", "description": "", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "u+random(0,z)", "description": "", "templateType": "anything"}, "c11": {"name": "c11", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "", "templateType": "anything"}, "ab21": {"name": "ab21", "group": "Ungrouped variables", "definition": "a21*b11+a22*b21+a23*b31", "description": "", "templateType": "anything"}, "v2": {"name": "v2", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "ba12": {"name": "ba12", "group": "Ungrouped variables", "definition": "b11*a12+b12*a22+b13*a32", "description": "", "templateType": "anything"}, "b32": {"name": "b32", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "b33": {"name": "b33", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "", "templateType": "anything"}, "s9": {"name": "s9", "group": "Ungrouped variables", "definition": "if(w=3,0.5,-0.5)", "description": "", "templateType": "anything"}, "s10": {"name": "s10", "group": "Ungrouped variables", "definition": "if(u=3,0.5,-0.5)", "description": "", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "m+random(0,z)", "description": "", "templateType": "anything"}, "ab22": {"name": "ab22", "group": "Ungrouped variables", "definition": "a21*b12+a22*b22+a23*b32", "description": "", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "templateType": "anything"}, "b22": {"name": "b22", "group": "Ungrouped variables", "definition": "random(-4,-3,-2,-1,1,2,3,4)", "description": "", "templateType": "anything"}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "a21*v1+a22*v2+a23*v3", "description": "", "templateType": "anything"}, "a31": {"name": "a31", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "u+random(0,z)", "description": "", "templateType": "anything"}, "c12": {"name": "c12", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "ab11": {"name": "ab11", "group": "Ungrouped variables", "definition": "a11*b11+a12*b21+a13*b31", "description": "", "templateType": "anything"}, "ba21": {"name": "ba21", "group": "Ungrouped variables", "definition": "b21*a11+b22*a21+b23*a31", "description": "", "templateType": "anything"}, "ba23": {"name": "ba23", "group": "Ungrouped variables", "definition": "b21*a13+b22*a23+b23*a33", "description": "", "templateType": "anything"}, "q3": {"name": "q3", "group": "Ungrouped variables", "definition": "b31*w1+b32*w2+b33*w3", "description": "", "templateType": "anything"}, "ab12": {"name": "ab12", "group": "Ungrouped variables", "definition": "a11*b12+a12*b22+a13*b32", "description": "", "templateType": "anything"}, "ba13": {"name": "ba13", "group": "Ungrouped variables", "definition": "b11*a13+b12*a23+b13*a33", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..6 except m)", "description": "", "templateType": "anything"}, "b31": {"name": "b31", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(4..6)", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "if(s=5,0.5,-0.5)", "description": "", "templateType": "anything"}, "s6": {"name": "s6", "group": "Ungrouped variables", "definition": "if(r=3,0.5,-0.5)", "description": "", "templateType": "anything"}, "ab31": {"name": "ab31", "group": "Ungrouped variables", "definition": "a31*b11+a32*b21+a33*b31", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(-1,1,2)", "description": "", "templateType": "anything"}, "b12": {"name": "b12", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything"}, "w3": {"name": "w3", "group": "Ungrouped variables", "definition": "random(-4..4)", "description": "", "templateType": "anything"}, "b13": {"name": "b13", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ba21", "a21", "a22", "a23", "b23", "b22", "b21", "ba22", "w3", "w2", "w1", "ba11", "a11", "ba13", "p1", "q1", "q", "q3", "q2", "ab23", "ab22", "ab21", "s3", "s2", "s1", "a12", "s7", "s6", "s5", "s4", "b12", "b13", "b11", "c13", "c12", "c11", "a33", "a32", "a31", "ba32", "ba33", "a13", "ba31", "s10", "s9", "b32", "b31", "v1", "v2", "v3", "mae", "mad", "mag", "mah", "ba12", "b33", "p2", "p3", "ba23", "ab11", "ab31", "ab32", "ab33", "ab12", "m", "s8", "n", "mac", "p", "s", "r", "u", "w", "v", "y", "x", "z", "ab13"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let</p>\n<p>\\[A = \\left(\\begin{array}{rrr} \\var{a11} &amp; \\var{a12} &amp; \\var{a13}\\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23}\\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33}\\\\ \\end{array}\\right),\\;\\;\\;\\; B= \\left(\\begin{array}{rrr} \\var{b11} &amp; \\var{b12} &amp; \\var{b13}\\\\ \\var{b21} &amp; \\var{b22} &amp; \\var{b23}\\\\ \\var{b31} &amp; \\var{b32} &amp; \\var{b33}\\\\ \\end{array}\\right),\\;\\;\\;\\; v= \\left(\\begin{array}{r} \\var{v1}\\\\ \\var{v2} \\\\ \\var{v3} \\end{array}\\right),\\;\\;\\;\\; w= \\left(\\begin{array}{r} \\var{w1}\\\\ \\var{w2} \\\\ \\var{w3} \\end{array}\\right)\\]</p>\n<p>Find the following products:</p>\n<p style=\"text-align: center;\">$Av= $&nbsp;[[0]]</p>\n<p style=\"text-align: center;\">$Bw= $&nbsp;[[1]]</p>\n<p style=\"text-align: center;\">$BA= $&nbsp;[[2]]</p>\n<p style=\"text-align: center;\">$AB= $&nbsp;[[3]]</p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Av", "marks": "1.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([[p1],[p2],[p3]])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": true, "customName": "Bw", "marks": "1.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([[q1],[q2],[q3]])", "correctAnswerFractions": false, "numRows": "3", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": true, "customName": "BA", "marks": "4.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([ba11,ba12,ba13],\n  [ba21,ba22,ba23],\n  [ba31,ba32,ba33])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": true, "customName": "AB", "marks": "4.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([ab11,ab12,ab13],\n  [ab21,ab22,ab23],\n  [ab31,ab32,ab33])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Consider the following matrices together with the matrices from the first part of the question.</p>\n        <p>\\[\\begin{eqnarray}&amp;C=&amp; \\var{mac},\\;\\;\\;\\; &amp;D=&amp; \\var{mad},\\;\\;\\; \\;&amp;E= &amp;\\var{mae}\\\\&amp;F=&amp; \\left(\\begin{array}{rr} \\var{w1} &amp; \\var{a12}\\\\ \\var{w2} &amp; \\var{b23} \\\\ \\var{w3} &amp; \\var{w2} \\\\\\var{v1} &amp; \\var{b12}\\\\ 0 &amp; \\var{-w2} \\end{array}\\right),\\;\\;\\;\\;&amp;G=&amp;\\var{mag},\\;\\;\\;\\;&amp;H=&amp;\\var{mah} \\end{eqnarray}\\]</p>\n        <p>Which of the following products of matrices can be calculated?</p>\n        <p>[[0]]</p>\n        <p><strong>Please note that for every correct answer you get 0.5 marks and for every incorrect answer 0.5 is taken away. The minimum mark you can get is 0.</strong></p>", "gaps": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "choices": ["<p>$CD$</p>", "<p>$DC$</p>", "<p>$EF$</p>", "<p>$FE$</p>", "<p>$BC$</p>", "<p>$AE$</p>", "<p>$GH$</p>", "<p>$HE$</p>", "<p>$AG$</p>", "<p>$GB$</p>"], "matrix": "v", "layout": {"type": "all", "expression": ""}, "answers": ["<strong>Can be calculated</strong>", "<strong>Cannot be calculated</strong>"]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Multiply 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([ [random(-3,-1,0,3),random(-3..1)], [random(2,3),random(-3..-1)] ])", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([ [random(-2,1,2),random(1..4)], [random(-2..2),random(1..3)] ])", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([ [random(1,0,4), a[0][1]+b[0][1]], [random(2..5),random(0,1)] ])", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "b", "c"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "a*b", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p style=\"text-align: center;\">$\\mathbf{AB} = \\var{A}\\var{B} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "b*a", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p style=\"text-align: center;\">$\\mathbf{BA} = \\var{B}\\var{A} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "c*b", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p style=\"text-align: center;\">$\\mathbf{CB} = \\var{C}\\var{B} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "a*c", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p style=\"text-align: center;\">$\\mathbf{AC} = \\var{A}\\var{C} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let</p>\n<p>\\begin{align} \\mathbf{A} &amp;= \\var{a}, &amp; \\mathbf{B} &amp;= \\var{b}, &amp; \\mathbf{C} &amp;= \\var{c} \\end{align}</p>\n<p>Calculate the following products of these matrices:</p>", "tags": ["checked2015", "MAS1602", "matrices", "matrix", "matrix multiplication", "matrix product", "multiplication of matrices", "multiplying matrices", "product of matrices", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>10/07/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>Display of matrices looks untidy when individual components include negative numbers.</p>\n        <p>Is it worthwhile restricting all components of matrices to be non zero?</p>\n        <p>Question appears to be working correctly.</p>\n        <p><strong>24/12/2012:</strong></p>\n        <p>Checked calculations, OK. Added tested1 tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Multiplication of $2 \\times 2$ matrices.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>\\begin{align}<br/>\\mathbf{AB} &amp;= \\var{A}\\var{B} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {a[0][0]}*{b[0][0]}+{a[0][1]}*{b[1][0]} } &amp; \\simplify[]{ {a[0][0]}*{b[0][1]} + {a[0][1]}*{b[1][1]} } \\\\ \\simplify[]{ {a[1][0]}*{b[0][0]} + {a[1][1]}*{b[1][0]} } &amp; \\simplify[]{ {a[1][0]}*{b[0][1]} + {a[1][1]}*{b[1][1]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{a*b}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>\\begin{align}<br/>\\mathbf{BA} &amp;= \\var{B}\\var{A} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {b[0][0]}*{a[0][0]}+{b[0][1]}*{a[1][0]} } &amp; \\simplify[]{ {b[0][0]}*{a[0][1]} + {b[0][1]}*{a[1][1]} } \\\\ \\simplify[]{ {b[1][0]}*{a[0][0]} + {b[1][1]}*{a[1][0]} } &amp; \\simplify[]{ {b[1][0]}*{a[0][1]} + {b[1][1]}*{a[1][1]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{b*a}<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>\\begin{align}<br/>\\mathbf{CB} &amp;= \\var{C}\\var{B} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {c[0][0]}*{b[0][0]}+{c[0][1]}*{b[1][0]} } &amp; \\simplify[]{ {c[0][0]}*{b[0][1]} + {c[0][1]}*{b[1][1]} } \\\\ \\simplify[]{ {c[1][0]}*{b[0][0]} + {c[1][1]}*{b[1][0]} } &amp; \\simplify[]{ {c[1][0]}*{b[0][1]} + {c[1][1]}*{b[1][1]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{c*b}<br/>\\end{align}</p>\n<h4>d)</h4>\n<p>\\begin{align}<br/>\\mathbf{AC} &amp;= \\var{A}\\var{C} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {a[0][0]}*{c[0][0]}+{a[0][1]}*{c[1][0]} } &amp; \\simplify[]{ {a[0][0]}*{c[0][1]} + {a[0][1]}*{c[1][1]} } \\\\ \\simplify[]{ {a[1][0]}*{c[0][0]} + {a[1][1]}*{c[1][0]} } &amp; \\simplify[]{ {a[1][0]}*{c[0][1]} + {a[1][1]}*{c[1][1]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{a*c}<br/>\\end{align}</p>"}, {"name": "Multiply 2x2, 2x1 and 1x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"B": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([random(1..4),random(-1,1,2)],[random(-1,0,1),random(-4,-3,-2,-1,1,2,3,5)])", "description": "", "name": "B"}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([[random(-5..5),random(-7..7 except 0)]])", "description": "", "name": "w"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([[random(-2..2)],[random(1..9)]])", "description": "", "name": "v"}}, "ungrouped_variables": ["B", "w", "v"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "w*v", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": 1, "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>$\\mathbf{w} \\; \\mathbf{v} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "w*B", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": 1, "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>$\\mathbf{w} \\mathbf{B} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "B*v", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>$\\mathbf{B} \\mathbf{v} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Answer the following questions on matrices.</p>\n<p>Let</p>\n<p>\\begin{align} \\mathbf{v} &amp;= \\var{v}, &amp; \\mathbf{B} &amp;= \\var{B}, &amp; \\mathbf{w} &amp;= \\var{w}&nbsp;\\end{align}</p>", "tags": ["checked2015", "MAS1602", "matrices", "matrix", "matrix multiplication", "matrix product", "multiplication of matrices", "multiply matrix", "product of matrices", "tested1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>10/07/2012:</strong></p>\n<p>Added tags.<strong><br /></strong></p>\n<p>Perhaps it would be worthwhile restraining all components of the three matrices to be non zero?</p>\n<p>Question appears to be working correctly.</p>\n<p><strong>24/12/2012:</strong></p>\n<p>Calculation checked, OK. Added tested1 tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Very elementary matrix multiplication.&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>\\begin{align} <br/>\\mathbf{w} \\mathbf{v} &amp;= \\simplify{{w}*{v}}&nbsp;\\\\ <br/>&amp;= \\simplify[]{{w[0][0]}*{v[0][0]}+{w[0][1]}*{v[1][0]}} \\\\ <br/>&amp;= \\var{w*v} <br/>\\end{align}</p>\n<h4>b)</h4>\n<p>Since $\\mathbf{w}$ is a $1 \\times 2$ matrix and has the same number of columns, $2$, as the rows in the $2 \\times 2$ matrix $\\mathbf{B}$ we can multiply and form&nbsp;$\\mathbf{w} \\mathbf{B}$ which is a $1 \\times 2$ row vector:</p>\n<p>\\begin{align}<br/>\\mathbf{wB} &amp;= \\simplify{{w}*{B}} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {w[0][0]}*{B[0][0]} + {w[0][1]}*{B[1][0]} }, &amp; \\simplify[]{ {w[0][0]}*{B[0][1]} + {w[0][1]}*{B[1][1]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{w*B}<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>Since $\\mathbf{B}$ is a $2 \\times 2$ matrix and has the same number of columns, $2$, as the rows in $\\mathbf{v}$ we can form the matrix $\\mathbf{Bv}$ which will be a $2 \\times 1$ column vector.</p>\n<p>We find this vector to be:</p>\n<p>\\begin{align}<br/>\\mathbf{Bv} &amp;= \\simplify{{B}*{v}} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {B[0][0]}*{v[0][0]} + {B[0][1]}*{v[1][0]} } \\\\ \\simplify[]{ {B[1][0]}*{v[0][0]} + {B[1][1]}*{v[1][0]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{B*v}<br/>\\end{align}</p>"}, {"name": "Solve simultaneous equations by finding inverse matrix, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["checked2015", "inverse of a matrix", "Linear equations", "linear equations", "linear equations in matrix form", "matrices", "matrix", "matrix equations", "matrix form", "matrix multiplication", "multiply matrices", "multiply matrix", "solving linear equations", "system of linear equations", "tested1"], "metadata": {"description": "<p>Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Rewrite the following system of equations as a matrix equation</p>\n<p>\\[ \\mathbf{Av} = \\mathbf{b} \\]</p>\n<p>for a matrix $\\mathbf{A}$ and column vectors $\\mathbf{v}$ and $\\mathbf{b}$.</p>\n<p>\\begin{align}<br/>\\simplify[std]{ {ma[0][0]}x + {ma[0][1]}y} &amp;= \\var{mb[0][0]} \\\\<br/>\\simplify[std]{ {ma[1][0]}x + {ma[1][1]}y} &amp;= \\var{mb[1][0]}<br/>\\end{align}</p>\n<p><em><span>Input all numbers as fractions or integers and not as decimals.</span></em></p>", "advice": "<h4>a)</h4>\n<p>The equations can be written in the matrix form</p>\n<p>\\[ \\var{ma}\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\var{mb} \\]</p>\n<h4><strong></strong>b)</h4>\n<p>$\\mathrm{det}(\\mathbf{A}) = \\simplify[]{ {ma[0][0]}*{ma[1][1]} - {ma[0][1]}*{ma[1][0]}} = \\var{det(ma)} \\neq 0$, so $\\mathbf{A}$ is invertible.</p>\n<p>\\[ \\mathbf{A}^{-1} = \\simplify[fractionnumbers]{{ma_inverse}} \\]</p>\n<h4>c)</h4>\n<p>We have&nbsp;</p>\n<p>\\begin{align}<br/>\\mathbf{A}^{-1}\\mathbf{b} &amp;= \\simplify[fractionnumbers]{{ma_inverse}*{mb}} \\\\<br/>&amp;= \\simplify[fractionnumbers]{{ma_inverse*mb}}<br/>\\end{align}</p>\n<h4>d)</h4>\n<p>Rearrange the equation $\\mathbf{Av}=\\mathbf{b}$ to make $\\mathbf{v}$ the subject:</p>\n<p>\\begin{align}<br/>\\mathbf{A}^{-1}\\mathbf{A}\\mathbf{v} &amp;= \\mathbf{A}^{-1}\\mathbf{b} \\\\<br/>\\mathbf{v} &amp;= \\mathbf{A}^{-1}\\mathbf{b} \\\\ \\\\<br/>\\end{align}</p>\n<p>Hence,</p>\n<p>\\[ \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\simplify[fractionnumbers]{{ma_inverse*mb}} \\]</p>\n<p>That is,</p>\n<p>\\begin{align}<br/>x &amp;= \\simplify[fractionnumbers]{{x}}, \\\\ \\\\<br/>y &amp;= \\simplify[fractionnumbers]{{y}}<br/>\\end{align}</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"mb": {"name": "mb", "group": "Ungrouped variables", "definition": "matrix([\n    [random(-9..9 except 0)],\n    [random(-9..9 except 0)]\n])", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(ma_inverse*mb)[0][0]", "description": "", "templateType": "anything", "can_override": false}, "ma": {"name": "ma", "group": "Ungrouped variables", "definition": "matrix([\n    [a00,a01],\n    [a10,a11]\n])", "description": "<p>Matrix A. a10 is picked so it's non-singular, and a11 is never $\\pm a01$.</p>\n<p>No entry is 0.</p>", "templateType": "anything", "can_override": false}, "a10": {"name": "a10", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a00,-a00,a00*a11/a01])", "description": "", "templateType": "anything", "can_override": false}, "ma_inverse": {"name": "ma_inverse", "group": "Ungrouped variables", "definition": "matrix([\n    [ma[1][1], -ma[0][1]],\n    [-ma[1][0], ma[0][0]]\n])/det(ma)", "description": "", "templateType": "anything", "can_override": false}, "a01": {"name": "a01", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "(ma_inverse*mb)[1][0]", "description": "", "templateType": "anything", "can_override": false}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a01,-a01])", "description": "", "templateType": "anything", "can_override": false}, "a00": {"name": "a00", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ma", "a00", "a01", "a10", "a11", "mb", "ma_inverse", "x", "y"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\">$\\mathbf{A} = $&nbsp;[[0]]</p>\n<div style=\"text-align: center;\">$\\mathbf{v} = $&nbsp;\n<div class=\"matrix-input\">\n  <div class=\"matrix-wrapper\">\n    <span class=\"left-bracket\"></span>\n    <table class=\"matrix\">\n      <tbody>\n        <tr>\n          <td>[[1]]</td>\n        </tr>\n        <tr>\n          <td>[[2]]</td>\n        </tr>\n      </tbody>\n    </table>\n    <span class=\"right-bracket\"></span>\n  </div>\n</div>\n<p style=\"text-align: center;\">$\\mathbf{b} = $ [[3]]</p></div>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "ma", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "y", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "y", "value": ""}]}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "mb", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the inverse of $\\mathbf{A}$.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "ma_inverse", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Now find $\\mathbf{A}^{-1}\\mathbf{b}$.</p>\n<p><span>$\\mathbf{A}^{-1}\\mathbf{b} = $&nbsp;[[0]]</span></p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "ma_inverse*mb", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, 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