// Numbas version: exam_results_page_options
{"name": "Find determinant of a 3x3 matrix 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", "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"], "name": "Find determinant of a 3x3 matrix by row reduction", "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": {}, "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>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>", "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/"}]}