// Numbas version: exam_results_page_options
{"name": "Find the Determinant of a 3 x 3 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"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}], "name": "Find the Determinant of a 3 x 3 Matrix", "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>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}