// Numbas version: exam_results_page_options
{"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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"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"], "name": "Find determinant of a 3x3 matrix with a free variable by row reduction", "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": {}, "extensions": [], "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>", "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/"}]}]}], "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/"}]}