// Numbas version: finer_feedback_settings
{"name": "Matrices", "metadata": {"description": "<p>Matrix addition, multiplication. Finding inverse. Determinants. Systems of equations.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Linear combinations of 2 x 2 matrices", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["a", "q1", "c", "b", "r1", "q", "p", "p1", "apb", "lcab", "lcabc"], "tags": ["addition of matrices", "linear algebra", "linear combination of matrices", "matrices", "matrix"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{A+B} &amp;=&amp;\\simplify[std]{{a}+{b}}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{a[0][0]}+{b[0][0]}}&amp;&nbsp;\\simplify[std]{{a[0][1]}+{b[0][1]}}\\\\&nbsp;&nbsp;\\simplify[std]{{a[1][0]}+{b[1][0]}}&amp;\\simplify[std]{{a[1][1]}+{b[1][1]}} \\end{pmatrix}\\\\ &amp;=&amp;\\simplify{{apb}}\\\\ &nbsp;\\end{eqnarray*} \\]</p>\n<h4>b)&nbsp;</h4>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{p}A+{q}B} &amp;=&amp;\\simplify[std]{{p}{a}+{q}{b}}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{p}*{a[0][0]}+{q}*{b[0][0]}}&amp;&nbsp;\\simplify[std]{{p}*{a[0][1]}+{q}*{b[0][1]}}\\\\&nbsp;&nbsp;\\simplify[std]{{p}*{a[1][0]}+{q}*{b[1][0]}}&amp;\\simplify[std]{{p}*{a[1][1]}+{q}*{b[1][1]}} \\end{pmatrix}\\\\ &amp;=&amp;\\simplify{{lcab}}\\\\ &nbsp;\\end{eqnarray*} \\]</p>\n<h4>c)</h4>\n<p>\\[ \\begin{eqnarray*} \\simplify[std]{{p1}A+{q1}B+{r1}C} &amp;=&amp;\\simplify[std]{{p1}{a}+{q1}{b}+{r1}{c}}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{p1}*{a[0][0]}+{q1}*{b[0][0]}+{r1}*{c[0][0]}}&amp;&nbsp;\\simplify[std]{{p1}*{a[0][1]}+{q1}*{b[0][1]}+{r1}*{c[0][1]}}\\\\&nbsp;&nbsp;\\simplify[std]{{p1}*{a[1][0]}+{q1}*{b[1][0]}+{r1}*{c[1][0]}}&amp;\\simplify[std]{{p1}*{a[1][1]}+{q1}*{b[1][1]}+{r1}*{c[1][1]}} \\end{pmatrix}\\\\ &amp;=&amp;\\simplify{{lcabc}}\\\\ &nbsp;\\end{eqnarray*} \\]</p>\n<p>&nbsp;</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noleadingminus"]}, "parts": [{"prompt": "<p>$\\mathrm{A}+\\mathrm{B} = \\simplify[std]{{a}+{b}} = $ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "apb", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "$\\simplify{{p}A+{q}B = {p}{a}+{q}{b}}=$ [[0]]", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "lcab", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$\\simplify{{p1}A+{q1}B+{r1}C = {p1}{a}+{q1}{b}+{r1}{c}}=$ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "lcabc", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Let&nbsp;<br/>\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]<br/>Calculate the following $2 \\times 2$ matrices:</p>\n<p>&nbsp;</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "matrix(repeat(repeat(random(-5..5 except 0),2),2))", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "q1": {"definition": "random(-6..6 except [0,1,-1,p1,q])", "templateType": "anything", "group": "Ungrouped variables", "name": "q1", "description": ""}, "c": {"definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0],b[0][0]]),2),2))", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "matrix(repeat(repeat(random(-5..5 except [0,a[0][0]]),2),2))", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "r1": {"definition": "random(-6..6 except [0,1,-1,p1,q1])", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "q": {"definition": "random(-6..6 except [0,1,-1,p])", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "p": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "p1": {"definition": "random(2..6 except p)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "apb": {"definition": "a+b", "templateType": "anything", "group": "Ungrouped variables", "name": "apb", "description": ""}, "lcab": {"definition": "p*a+q*b", "templateType": "anything", "group": "Ungrouped variables", "name": "lcab", "description": ""}, "lcabc": {"definition": "p1*a+q1*b+r1*c", "templateType": "anything", "group": "Ungrouped variables", "name": "lcabc", "description": ""}}, "metadata": {"notes": "\n            \t\t<p><strong>8/02/2013:</strong></p>\n            \t\t        <p><strong><br /></strong>Finished first draft.</p>\n            \t\t", "description": "<p>Linear combinations of $2 \\times 2$ matrices. Three examples.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Matrix Multiplication 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["ba21", "a21", "a22", "ba22", "cb21", "b22", "b21", "cb22", "ac22", "ac21", "ab22", "ab21", "b12", "b11", "c12", "c11", "c22", "a11", "cb11", "cb12", "a12", "c21", "ba11", "ba12", "ab12", "ab11", "ac12", "ac11"], "tags": ["matrices", "matrix", "matrix multiplication", "matrix product", "multiplication of matrices", "multiplying matrices", "product of matrices"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>\\[ \\begin{eqnarray*} AB &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{b11}+{a12}{b21}}&amp;\\simplify[]{{a11}{b12}+{a12}{b22}}\\\\ \\simplify[]{{a21}{b11}+{a22}{b21}}&amp;\\simplify[]{{a21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ab11}&amp;\\var{ab12}\\\\ \\var{ab21}&amp;\\var{ab22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>b)</h4>\n<p>\\[ \\begin{eqnarray*} BA &amp;=&amp; \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{b11}{a11}+{b12}{a21}}&amp;\\simplify[]{{b11}{a12}+{b12}{a22}}\\\\ \\simplify[]{{b21}{a11}+{b22}{a21}}&amp;\\simplify[]{{b21}{a12}+{b22}{a22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ba11}&amp;\\var{ba12}\\\\ \\var{ba21}&amp;\\var{ba22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>c)</h4>\n<p>\\[ \\begin{eqnarray*} CB &amp;=&amp; \\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{c11}{b11}+{c12}{b21}}&amp;\\simplify[]{{c11}{b12}+{c12}{b22}}\\\\ \\simplify[]{{c21}{b11}+{c22}{b21}}&amp;\\simplify[]{{c21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{cb11}&amp;\\var{cb12}\\\\ \\var{cb21}&amp;\\var{cb22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<h4>d)</h4>\n<p>\\[ \\begin{eqnarray*} AC &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{c11}+{a12}{c21}}&amp;\\simplify[]{{a11}{c12}+{a12}{c22}}\\\\ \\simplify[]{{a21}{c11}+{a22}{c21}}&amp;\\simplify[]{{a21}{c12}+{a22}{c22}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ac11}&amp;\\var{ac12}\\\\ \\var{ac21}&amp;\\var{ac22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "parts": [{"prompt": "<p>$AB = \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix} = $ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ab11,ab12],\n    [ab21,ab22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$BA = \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}=$ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ba11,ba12],\n    [ba21,ba22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$CB = \\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix} \\begin{pmatrix} \\var{b11}&amp;\\var{b12}\\\\ \\var{b21}&amp;\\var{b22}\\\\ \\end{pmatrix}=$&nbsp;[[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [cb11,cb12],\n    [cb21,cb22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$AC = \\begin{pmatrix} \\var{a11}&amp;\\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&amp;\\var{c12}\\\\ \\var{c21}&amp;\\var{c22}\\\\ \\end{pmatrix}=$&nbsp;[[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": false, "correctAnswerFractions": false, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "matrix([\n    [ac11,ac12],\n    [ac21,ac22]\n])", "scripts": {}, "marks": 1, "tolerance": 0, "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n  \n  \n  <p>Do the following matrix problems <br/>Let <br/>\\[A=\\begin{pmatrix} \\var{a11}&#38;\\var{a12}\\\\ \\var{a21}&#38;\\var{a22}\\\\ \\end{pmatrix},\\;\\;\n  \n  B=\\begin{pmatrix} \\var{b11}&#38;\\var{b12}\\\\ \\var{b21}&#38;\\var{b22}\\\\ \\end{pmatrix},\\;\\;\n  \n  C=\\begin{pmatrix} \\var{c11}&#38;\\var{c12}\\\\ \\var{c21}&#38;\\var{c22}\\\\ \\end{pmatrix}\\]<br/>Calculate the following products of these matrices:</p>\n  \n  \n    \n    ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ba21": {"definition": "b21*a11+b22*a21", "templateType": "anything", "group": "Ungrouped variables", "name": "ba21", "description": ""}, "a21": {"definition": "random(-2..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a21", "description": ""}, "a22": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "a22", "description": ""}, "ba22": {"definition": "b21*a12+b22*a22", "templateType": "anything", "group": "Ungrouped variables", "name": "ba22", "description": ""}, "cb21": {"definition": "c21*b11+c22*b21", "templateType": "anything", "group": "Ungrouped variables", "name": "cb21", "description": ""}, "b22": {"definition": "random(-3..-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b22", "description": ""}, "b21": {"definition": "random(2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b21", "description": ""}, "cb22": {"definition": "c21*b12+c22*b22", "templateType": "anything", "group": "Ungrouped variables", "name": "cb22", "description": ""}, "ac22": {"definition": "a21*c12+a22*c22", "templateType": "anything", "group": "Ungrouped variables", "name": "ac22", "description": ""}, "ac21": {"definition": "a21*c11+a22*c21", "templateType": "anything", "group": "Ungrouped variables", "name": "ac21", "description": ""}, "ab22": {"definition": "a21*b12+a22*b22", "templateType": "anything", "group": "Ungrouped variables", "name": "ab22", "description": ""}, "ab21": {"definition": "a21*b11+a22*b21", "templateType": "anything", "group": "Ungrouped variables", "name": "ab21", "description": ""}, "b12": {"definition": "random(-3..1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b12", "description": ""}, "b11": {"definition": "random(-3,-1,0,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b11", "description": ""}, "c12": {"definition": "a12+b12", "templateType": "anything", "group": "Ungrouped variables", "name": "c12", "description": ""}, "c11": {"definition": "random(1,0,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c11", "description": ""}, "ab12": {"definition": "a11*b12+a12*b22", "templateType": "anything", "group": "Ungrouped variables", "name": "ab12", "description": ""}, "a11": {"definition": "random(-2,1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a11", "description": ""}, "cb11": {"definition": "c11*b11+c12*b21", "templateType": "anything", "group": "Ungrouped variables", "name": "cb11", "description": ""}, "cb12": {"definition": "c11*b12+c12*b22", "templateType": "anything", "group": "Ungrouped variables", "name": "cb12", "description": ""}, "a12": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a12", "description": ""}, "ab11": {"definition": "a11*b11+a12*b21", "templateType": "anything", "group": "Ungrouped variables", "name": "ab11", "description": ""}, "ba11": {"definition": "b11*a11+b12*a21", "templateType": "anything", "group": "Ungrouped variables", "name": "ba11", "description": ""}, "ba12": {"definition": "b11*a12+b12*a22", "templateType": "anything", "group": "Ungrouped variables", "name": "ba12", "description": ""}, "c22": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c22", "description": ""}, "c21": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c21", "description": ""}, "ac12": {"definition": "a11*c12+a12*c22", "templateType": "anything", "group": "Ungrouped variables", "name": "ac12", "description": ""}, "ac11": {"definition": "a11*c11+a12*c21", "templateType": "anything", "group": "Ungrouped variables", "name": "ac11", "description": ""}}, "metadata": {"notes": "\n        \t\t                \t\t<p><strong>10/07/2012:</strong></p>\n        \t\t                \t\t<p>Added tags.</p>\n        \t\t                \t\t<p>Display of matrices looks untidy when individual components include negative numbers.</p>\n        \t\t                \t\t<p>Is it worthwhile restricting all components of matrices to be non zero?</p>\n        \t\t                \t\t<p>Question appears to be working correctly.</p>\n        \t\t                \n        \t\t", "description": "<p>Multiplication of $2 \\times 2$ matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Determinants ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "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"], "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 the determinant of a $2 \\times 2$ matrix:<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>Calculate the determinants of these matrices:</p>\n<p>$\\mathrm{det}\\left(A\\right) = \\;\\;$[[0]]$,\\;\\;\\;\\mathrm{det}\\left(B\\right) = \\;\\;$[[1]]$,\\;\\;\\;\\mathrm{det}\\left(C\\right) = \\;\\;$[[2]]$,\\;\\;\\;\\mathrm{det}\\left(ABC\\right) = \\;\\;$[[3]]</p>", "marks": 0, "gaps": [{"allowFractions": false, "scripts": {}, "maxValue": "dA", "minValue": "dA", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "dB", "minValue": "dB", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "dC", "minValue": "dC", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "dA*dB*dC", "minValue": "dA*dB*dC", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": 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>", "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 determinant of three $2 \\times 2$ invertible matrices.</p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Determinant and inverse of 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {"inverse": {"definition": "matrix([\n    [m[1][1],-m[0][1]],\n    [-m[1][0],m[0][0]]\n])/det(m)", "type": "matrix", "language": "jme", "parameters": [["m", "matrix"]]}}, "ungrouped_variables": [], "tags": ["determinant of a matrix", "inverse", "inverse matrix", "matrices", "matrix", "matrix inverse", "matrix multiplication", "multiplication of matrices", "tested1"], "preamble": {"css": "", "js": ""}, "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>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "parts": [{"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>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.5, "maxValue": "det(a)", "minValue": "det(a)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "det(b)", "minValue": "det(b)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "det(c)", "minValue": "det(c)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "det(a*b*c)", "minValue": "det(a*b*c)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": true, "correctAnswerFractions": true, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "inverse(a)", "scripts": {}, "marks": 1, "tolerance": "0.01", "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$\\mathrm{B}^{-1} = $ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": true, "correctAnswerFractions": true, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "inverse(b)", "scripts": {}, "marks": 1, "tolerance": "0.01", "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$\\mathrm{C}^{-1} = $ [[0]]</p>", "marks": 0, "gaps": [{"numColumns": "2", "type": "matrix", "allowFractions": true, "correctAnswerFractions": true, "markPerCell": false, "numRows": "2", "showCorrectAnswer": true, "correctAnswer": "inverse(c)", "scripts": {}, "marks": 1, "tolerance": "0.01", "allowResize": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Do the following matrix problems.</p>", "variable_groups": [{"variables": ["a11", "a12", "a21", "a22", "a"], "name": "Matrix A"}, {"variables": ["b11", "b12", "b21", "b22", "b"], "name": "Matrix B"}, {"variables": ["c11", "c12", "c21", "c22", "c"], "name": "Matrix C"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"c22": {"definition": "random(1..9 except c21*c12/c11)", "templateType": "anything", "group": "Matrix C", "name": "c22", "description": ""}, "a21": {"definition": "random(-6..6 except 0) ", "templateType": "anything", "group": "Matrix A", "name": "a21", "description": ""}, "a22": {"definition": "random(1..9 except a21*a12/a11)", "templateType": "anything", "group": "Matrix A", "name": "a22", "description": ""}, "c21": {"definition": "random(2..5)", "templateType": "anything", "group": "Matrix C", "name": "c21", "description": ""}, "a11": {"definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Matrix A", "name": "a11", "description": ""}, "a12": {"definition": "random(-5..5)", "templateType": "anything", "group": "Matrix A", "name": "a12", "description": ""}, "c": {"definition": "matrix([\n    [c11,c12],\n    [c21,c22]\n])", "templateType": "anything", "group": "Matrix C", "name": "c", "description": ""}, "b21": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Matrix B", "name": "b21", "description": ""}, "b22": {"definition": "random(-9..9 except [0,b21*b12/b11])", "templateType": "anything", "group": "Matrix B", "name": "b22", "description": ""}, "b": {"definition": "matrix([\n    [b11,b12],\n    [b21,b22]\n])", "templateType": "anything", "group": "Matrix B", "name": "b", "description": ""}, "b12": {"definition": "random(-5..5)", "templateType": "anything", "group": "Matrix B", "name": "b12", "description": ""}, "a": {"definition": "matrix([\n    [a11,a12],\n    [a21,a22]\n])", "templateType": "anything", "group": "Matrix A", "name": "a", "description": ""}, "b11": {"definition": "random(1..9 except a11)\n//if(a11=tr2,tr2+1,tr2)", "templateType": "anything", "group": "Matrix B", "name": "b11", "description": ""}, "c12": {"definition": "a12+b12", "templateType": "anything", "group": "Matrix C", "name": "c12", "description": ""}, "c11": {"definition": "random(1,2,4)", "templateType": "anything", "group": "Matrix C", "name": "c11", "description": ""}}, "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"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "cormac's copy of Matrix question", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}], "functions": {}, "tags": ["inverse of a matrix", "linear equations", "linear equations in matrix form", "matrices", "matrix", "matrix equations", "matrix form", "matrix multiplication", "multiply matrices", "multiply matrix", "solving linear equations", "system of linear equations"], "advice": "\n    <p>a)<br/>The equations can be written in the matrix form:<br/>\\[\\begin{pmatrix} \\var{a} &amp; \\var{b}\\\\ \\var{a1}&amp;\\var{b1} \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} \\var{c} \\\\ \\var{c1} \\end{pmatrix}\\]</p>\n    <p>b)</p>\n    <p>Since $\\mathrm{det}(A) = \\simplify[]{{a}*{b1}-{b}*{a1}={dA}} \\neq 0$, $A$ is invertible and</p>\n    <p>\\[A^{-1} = \\begin{pmatrix} \\simplify[std]{{b1}/{dA}}&amp;\\simplify[std]{{-b}/{dA}}\\\\\\simplify[std]{{-a1}/{dA}}&amp;\\simplify[std]{{a}/{dA}} \\end{pmatrix}\\]</p>\n    <p>c)</p>\n    <p>We have:<br/>\\[ \\begin{eqnarray*} A^{-1}b &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{b1}/{dA}}&amp;\\simplify[std]{{-b}/{dA}}\\\\\\simplify[std]{{-a1}/{dA}}&amp;\\simplify[std]{{a}/{dA}} \\end{pmatrix}\\begin{pmatrix} \\var{c}\\\\\\var{c1}\\end{pmatrix} \\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\\\simplify[std]{{c1*a-c*a1}/{dA}}\\end{pmatrix} \\end{eqnarray*} \\]</p>\n    <p>d) Note that $Av = b \\Rightarrow v = A^{-1}b$ hence we can read the solution from the last part as this gives:</p>\n    <p>\\[\\begin{pmatrix} x\\\\y \\end{pmatrix} = \\begin{pmatrix} \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\ \\simplify[std]{{c1*a-c*a1}/{dA}}\\end{pmatrix}\\]<br/>Hence \\[\\begin{eqnarray*} x&amp;=&amp; \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\ y&amp;=&amp; \\simplify[std]{{c1*a-c*a1}/{dA}} \\end{eqnarray*} \\]</p>\n    \n    ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <table>\n      <tbody>\n      <tr>\n      <td rowspan=\"2\">$A = \\Bigg($</td>\n      <td>[[0]]</td>\n      <td>[[1]]</td>\n      <td rowspan=\"2\">$\\Bigg)$</td>\n      </tr>\n      <tr>\n      <td>[[2]]</td>\n      <td>[[3]]</td>\n      </tr>\n      <tr>\n      <td rowspan=\"2\">$v = \\;\\;\\Bigg($</td>\n      <td>[[4]]</td>\n      <td rowspan=\"2\">$\\Bigg)$</td>\n      </tr>\n      <tr>\n      <td>[[5]]</td>\n      </tr>\n      <tr>\n      <td rowspan=\"2\">$b = \\;\\;\\Bigg($</td>\n      <td>[[6]]</td>\n      <td rowspan=\"2\">$\\Bigg)$</td>\n      </tr>\n      <tr>\n      <td>[[7]]</td>\n      </tr>\n      </tbody>\n      </table>\n        \n        ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "{a}", "type": "jme"}, {"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "{b}", "type": "jme"}, {"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "{a1}", "type": "jme"}, {"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "{b1}", "type": "jme"}, {"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "x", "type": "jme"}, {"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "y", "type": "jme"}, {"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "{c}", "type": "jme"}, {"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.25, "answer": "{c1}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n        <p>Find the inverse of $A$, input all numbers as fractions or integers and not as decimals:</p>\n      <table>\n      <tbody>\n      <tr>\n      <td rowspan=\"2\">$A^{-1} = \\Bigg($</td>\n      <td>[[0]]</td>\n      <td>[[1]]</td>\n      <td rowspan=\"2\">$\\Bigg)$</td>\n      </tr>\n      <tr>\n      <td>[[2]]</td>\n      <td>[[3]]</td>\n      </tr>\n      </tbody>\n      </table>\n        \n        ", "gaps": [{"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{b1}/{dA}", "type": "jme"}, {"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{-b}/{dA}", "type": "jme"}, {"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{-a1}/{dA}", "type": "jme"}, {"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{a}/{dA}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n        <p>Now find, inputting all numbers as fractions or integers and not as decimals.</p>\n      <table>\n      <tbody>\n      <tr>\n      <td rowspan=\"2\">$A^{-1}b = \\;\\;\\Bigg($</td>\n      <td>[[0]]</td>\n      <td rowspan=\"2\">$\\Bigg)$</td>\n      </tr>\n      <tr>\n      <td>[[1]]</td>\n      </tr>\n      </tbody>\n      </table>\n        \n        ", "gaps": [{"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{c*b1-c1*b}/{b1*a-a1*b}", "type": "jme"}, {"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{-c*a1+c1*a}/{b1*a-a1*b}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n        \n        \n        <p>Now solve the equations, inputting all numbers as fractions or integers and not as decimals.<br/>$x = \\;\\;$[[0]]</p>\n        \n        \n        \n        <p>$y = \\;\\;$[[1]]</p>\n        \n        \n        \n        ", "gaps": [{"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{c*b1-c1*b}/{b1*a-a1*b}", "type": "jme"}, {"notallowed": {"message": "<p>Input as a fraction or an integer, not as a decimal</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 0.5, "answer": "{-c*a1+c1*a}/{b1*a-a1*b}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Write the following equations as a matrix equation <br/>\\[Av=b\\]for a matrix $A$ and column vectors $v$ and $b$<br/>\\[ \\begin{eqnarray*} \\simplify[std]{{a}x+{b}y}&amp;=&amp;\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&amp;=&amp;\\var{c1} \\end{eqnarray*} \\]</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "sa*random(1..9)", "name": "a"}, "addortakeaway": {"definition": "if(b*b1<0,'add the equation','take away the equation')", "name": "addortakeaway"}, "c": {"definition": "sc*random(1..9)", "name": "c"}, "b": {"definition": "sb*random(1..9)", "name": "b"}, "da": {"definition": "a*b1-a1*b", "name": "da"}, "a1": {"definition": "if(test+inc=0,test+inc+1,test+inc)", "name": "a1"}, "test": {"definition": "round(a*b1/b)", "name": "test"}, "ab": {"definition": "abs(b)", "name": "ab"}, "sb1": {"definition": "random(1,-1)", "name": "sb1"}, "b1": {"definition": "if(ab=1,sb1*random(2..9),if(ab=2 or ab=4 or ab=8,sb1*random(1,3,5,7,9), if(ab=3 or ab=6 or ab=9,sb1*random(1,2,4,5,6,8),if(ab=5,sb1*random(1,2,3,4,6,7,8,9),sb1*random(1,2,3,4,5,6,8,9)))))", "name": "b1"}, "sc": {"definition": "random(1,-1)", "name": "sc"}, "sb": {"definition": "random(1,-1)", "name": "sb"}, "sa": {"definition": "random(1,-1)", "name": "sa"}, "sc1": {"definition": "random(1,-1)", "name": "sc1"}, "sa1": {"definition": "random(1,-1)", "name": "sa1"}, "c1": {"definition": "sc1*random(1..9)", "name": "c1"}, "inc": {"definition": "sa*random(1..9)", "name": "inc"}}, "metadata": {"notes": "\n        \t\t        \t\t<p><strong>20/06/2012:</strong></p>\n        \t\t        \t\t<p>Added, edited tags.</p>\n        \t\t        \t\t<p>Edited advice so that it gave the correct solution for $y$ (as in the answer).</p>\n        \t\t        \t\t<p>&nbsp;</p>\n        \t\t        \t\t<p>&nbsp;</p>\n        \t\t        \t\t<p><strong>4/07/2012:<br /></strong></p>\n        \t\t        \t\t<p>Column vectors v and b have the bracket in the incorrect place.</p>\n        \t\t        \t\t<p>&nbsp;</p>\n        \t\t        \t\t<p><strong>10/07/2012:<br /></strong></p>\n        \t\t        \t\t<p>Added tags.<strong><br /></strong></p>\n        \t\t        \t\t<p>Question appears to be working correctly.</p>\n        \t\t        \t\t<p>Column vectors v and b still have brackets in incorrect places.</p>\n        \t\t        \n        \t\t", "description": "<p>Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.&nbsp;</p>"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": false, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, 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