// Numbas version: exam_results_page_options
{"name": "2x2 Inverses", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a21", "a22", "b22", "b21", "b1", "s2", "s1", "b12", "b11", "tr1", "c12", "c11", "tr2", "tr4", "c22", "a11", "a12", "db", "dc", "da", "a1", "c21", "c1", "tr3", "a", "b", "s", "u", "t"], "name": "2x2 Inverses", "tags": ["determinant of a matrix", "inverse", "inverse matrix", "matrix", "matrix inverse", "matrix multiplication", "multiply matrix"], "advice": "<h4>a)</h4>\n<h5>Determinants.</h5>\n<p>Here is the formula for determinants:<br/>\\[M = \\begin{pmatrix} a &amp; b \\\\ c&amp;d \\end{pmatrix} \\Rightarrow \\mathrm{det}\\left(M\\right) = ad-bc \\]</p>\n<p>$\\mathrm{det}\\left(A\\right) = \\simplify[]{{a11}*{a22}-{a12}*{a21} = {dA}}$<br/>$\\mathrm{det}\\left(B\\right) = \\simplify[]{{b11}*{b22}-{b12}*{b21} = {dB}}$<br/>$\\mathrm{det}\\left(C\\right) = \\simplify[]{{c11}*{c22}-{c12}*{c21} = {dC}}$</p>\n<h5>Determinant of a product of matrices.</h5>\n<p>If we have two $n \\times n$ matrices $M$ and $N$ then:<br/>\\[\\mathrm{det}\\left(MN\\right) = \\mathrm{det}\\left(M\\right)\\mathrm{det}\\left(N\\right)\\]<br/>And it follows that if we have a third matrix $P$ that:<br/>\\[\\mathrm{det}\\left(MNP\\right) = \\mathrm{det}\\left(M\\right)\\mathrm{det}\\left(N\\right)\\mathrm{det}\\left(P\\right)\\]</p>\n<p>Thus for our example we have:</p>\n<p>\\[\\begin{eqnarray*}\\mathrm{det}\\left(ABC\\right) &amp;=&amp; \\mathrm{det}\\left(A\\right)\\times\\mathrm{det}\\left(B\\right)\\times\\mathrm{det}\\left(C\\right)\\\\ &amp;=&amp; \\var{dA}\\times \\var{dB} \\times \\var{dC}\\\\ &amp;=&amp; \\var{dA*dB*dC} \\end{eqnarray*} \\]</p>\n<h5>Inverse of a $2 \\times 2$ matrix</h5>\n<p>Suppose $M$ is a $2 \\times 2$ matrix and $\\mathrm{det}\\left(M\\right) = \\Delta \\neq 0$.</p>\n<p>Then $M$ is invertible and:<br/>\\[M = \\begin{pmatrix} a &amp; b \\\\ c&amp;d \\end{pmatrix} \\Rightarrow 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>\\[A^{-1} = \\begin{pmatrix} \\simplify[std]{{a22}/{dA}} &amp;\\simplify[std]{{-a12}/{dA}}\\\\\\simplify[std]{{-a21}/{dA}}&amp;\\simplify[std]{{a11}/{dA}}\\end{pmatrix}\\]</p>\n<h4>c)</h4>\n<p>\\[B^{-1} = \\begin{pmatrix} \\simplify[std]{{b22}/{dB}} &amp;\\simplify[std]{{-b12}/{dB}}\\\\\\simplify[std]{{-b21}/{dB}}&amp;\\simplify[std]{{b11}/{dB}}\\end{pmatrix}\\]</p>\n<h4>d)</h4>\n<p>\\[C^{-1} = \\begin{pmatrix} \\simplify[std]{{c22}/{dC}} &amp;\\simplify[std]{{-c12}/{dC}}\\\\\\simplify[std]{{-c21}/{dC}}&amp;\\simplify[std]{{c11}/{dC}}\\end{pmatrix}\\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "parts": [{"prompt": "<p>$A^{-1} = $ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "tolerance": 0, "allowFractions": true, "scripts": {}, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [a22,-a12],\n    [-a21,a11]\n])/dA", "correctAnswerFractions": true, "marks": 1, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$B^{-1} = $ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "tolerance": 0, "allowFractions": true, "scripts": {}, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [b22,-b12],\n    [-b21,b11]\n])/dB", "correctAnswerFractions": true, "marks": 1, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$C^{-1} = $ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "tolerance": 0, "allowFractions": true, "scripts": {}, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [c22,-c12],\n    [-c21,c11]\n])/dB", "correctAnswerFractions": true, "marks": 1, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Let <br/>\\[A=\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix},\\;\\; B=\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix},\\;\\; C=\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\]</p>\n<p>Find the following matrix inverses. Input all matrix entries as fractions or integers and not as decimals.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a21": {"definition": "s2*random(1..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "a21", "description": ""}, "a22": {"definition": "if(tr1*a11=a21*a12,tr1+1,tr1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a22", "description": ""}, "b22": {"definition": "if(tr3*b11=b21*b12,tr3+1,tr3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b22", "description": ""}, "b21": {"definition": "s*random(1..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "b21", "description": ""}, "b1": {"definition": "t*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "b12": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b12", "description": ""}, "b11": {"definition": "if(a11=tr2,tr2+1,tr2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b11", "description": ""}, "tr1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "tr1", "description": ""}, "c12": {"definition": "a12+b12", "templateType": "anything", "group": "Ungrouped variables", "name": "c12", "description": ""}, "c11": {"definition": "random(1,2,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c11", "description": ""}, "tr2": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "tr2", "description": ""}, "tr4": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "tr4", "description": ""}, "c22": {"definition": "if(tr4*c11=c21*c12,tr4+1,tr4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c22", "description": ""}, "a11": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a11", "description": ""}, "a12": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a12", "description": ""}, "db": {"definition": "b11*b22-b21*b12", "templateType": "anything", "group": "Ungrouped variables", "name": "db", "description": ""}, "dc": {"definition": "c11*c22-c21*c12", "templateType": "anything", "group": "Ungrouped variables", "name": "dc", "description": ""}, "da": {"definition": "a11*a22-a21*a12", "templateType": "anything", "group": "Ungrouped variables", "name": "da", "description": ""}, "a1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "c21": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c21", "description": ""}, "c1": {"definition": "u*random(1..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "tr3": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "tr3", "description": ""}, "a": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "s*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "s": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "u": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "t": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}}, "metadata": {"notes": "\n    \t\t    \t\t        \t\t<p><strong>10/07/2012:</strong></p>\n    \t\t    \t\t        \t\t<p>Added tags.<strong><br /></strong></p>\n    \t\t    \t\t        \t\t<p>Question appears to be working correctly.</p>\n    \t\t    \t\t        \t\t<p>Corrected a typo in the Advice section.<strong><br /></strong></p>\n    \t\t    \t\t        \n    \t\t    \n    \t\t", "description": "<p>Find the inverse of three $2 \\times 2$ invertible matrices.</p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hayley Bishop", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/93/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hayley Bishop", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/93/"}]}