// Numbas version: finer_feedback_settings
{"name": "Ch10 Q3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ch10 Q3", "ungrouped_variables": [], "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>", "description": "<p>Find the determinant and inverse of three $2 \\times 2$ invertible matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "preamble": {"js": "", "css": ""}, "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>", "variables": {"b22": {"name": "b22", "definition": "random(-9..9 except [0,b21*b12/b11])", "templateType": "anything", "group": "Matrix B", "description": ""}, "b11": {"name": "b11", "definition": "random(1..9 except a11)\n//if(a11=tr2,tr2+1,tr2)", "templateType": "anything", "group": "Matrix B", "description": ""}, "b12": {"name": "b12", "definition": "random(-5..5)", "templateType": "anything", "group": "Matrix B", "description": ""}, "c22": {"name": "c22", "definition": "random(1..9 except c21*c12/c11)", "templateType": "anything", "group": "Matrix C", "description": ""}, "a22": {"name": "a22", "definition": "random(1..9 except a21*a12/a11)", "templateType": "anything", "group": "Matrix A", "description": ""}, "a11": {"name": "a11", "definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Matrix A", "description": ""}, "c11": {"name": "c11", "definition": "random(1,2,4)", "templateType": "anything", "group": "Matrix C", "description": ""}, "c": {"name": "c", "definition": "matrix([\n    [c11,c12],\n    [c21,c22]\n])", "templateType": "anything", "group": "Matrix C", "description": ""}, "c12": {"name": "c12", "definition": "a12+b12", "templateType": "anything", "group": "Matrix C", "description": ""}, "b21": {"name": "b21", "definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Matrix B", "description": ""}, "c21": {"name": "c21", "definition": "random(2..5)", "templateType": "anything", "group": "Matrix C", "description": ""}, "b": {"name": "b", "definition": "matrix([\n    [b11,b12],\n    [b21,b22]\n])", "templateType": "anything", "group": "Matrix B", "description": ""}, "a": {"name": "a", "definition": "matrix([\n    [a11,a12],\n    [a21,a22]\n])", "templateType": "anything", "group": "Matrix A", "description": ""}, "a12": {"name": "a12", "definition": "random(-5..5)", "templateType": "anything", "group": "Matrix A", "description": ""}, "a21": {"name": "a21", "definition": "random(-6..6 except 0) ", "templateType": "anything", "group": "Matrix A", "description": ""}}, "variable_groups": [{"name": "Matrix A", "variables": ["a11", "a12", "a21", "a22", "a"]}, {"name": "Matrix B", "variables": ["b11", "b12", "b21", "b22", "b"]}, {"name": "Matrix C", "variables": ["c11", "c12", "c21", "c22", "c"]}], "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "functions": {"inverse": {"definition": "matrix([\n    [m[1][1],-m[0][1]],\n    [-m[1][0],m[0][0]]\n])/det(m)", "type": "matrix", "parameters": [["m", "matrix"]], "language": "jme"}}, "statement": "<p>Do the following matrix problems.</p>", "tags": ["determinant of a matrix", "inverse", "inverse matrix", "matrices", "matrix", "matrix inverse", "matrix multiplication", "multiplication of matrices", "tested1"], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "parts": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "maxValue": "det(a)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "scripts": {}, "minValue": "det(a)", "showPrecisionHint": false}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "maxValue": "det(b)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "scripts": {}, "minValue": "det(b)", "showPrecisionHint": false}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "maxValue": "det(c)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "scripts": {}, "minValue": "det(c)", "showPrecisionHint": false}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "maxValue": "det(a*b*c)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "scripts": {}, "minValue": "det(a*b*c)", "showPrecisionHint": false}], "marks": 0, "scripts": {}, "prompt": "<p>Let</p>\n<p>\\[\\mathrm{A} = \\var{a},\\;\\; \\mathrm{B} = \\var{b},\\;\\; \\mathrm{C} = \\var{c}\\]</p>\n<p>Calculate the determinants of these matrices:</p>\n<p>$\\det\\left(\\mathrm{A}\\right) = $ [[0]]</p>\n<p>$\\det\\left(\\mathrm{B}\\right) = $ [[1]]</p>\n<p>$\\det\\left(\\mathrm{C}\\right) = $ [[2]]</p>\n<p>$\\det\\left(\\mathrm{ABC}\\right) = $ [[3]]</p>", "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "matrix", "numColumns": "2", "allowFractions": true, "tolerance": "0.01", "correctAnswer": "inverse(a)", "scripts": {}, "markPerCell": false, "correctAnswerFractions": true, "marks": 1, "allowResize": false, "numRows": "2", "showCorrectAnswer": true}], "marks": 0, "scripts": {}, "prompt": "<p>Find the inverses of the matrices given above. Input all matrix entries as fractions or integers and not as decimals.</p>\n<p>$\\mathrm{A}^{-1} = $ [[0]]</p>", "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "matrix", "numColumns": "2", "allowFractions": true, "tolerance": "0.01", "correctAnswer": "inverse(b)", "scripts": {}, "markPerCell": false, "correctAnswerFractions": true, "marks": 1, "allowResize": false, "numRows": "2", "showCorrectAnswer": true}], "marks": 0, "scripts": {}, "prompt": "<p>$\\mathrm{B}^{-1} = $ [[0]]</p>", "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "matrix", "numColumns": "2", "allowFractions": true, "tolerance": "0.01", "correctAnswer": "inverse(c)", "scripts": {}, "markPerCell": false, "correctAnswerFractions": true, "marks": 1, "allowResize": false, "numRows": "2", "showCorrectAnswer": true}], "marks": 0, "scripts": {}, "prompt": "<p>$\\mathrm{C}^{-1} = $ [[0]]</p>", "showCorrectAnswer": true}], "contributors": [{"name": "Alex Van den Hof", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1273/"}]}]}], "contributors": [{"name": "Alex Van den Hof", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1273/"}]}