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Systems of equations.\u003c/p\u003e\\n\u003cp\u003erebelmaths\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"duration\": 1860, \"percentPass\": 0, \"showQuestionGroupNames\": false, \"showstudentname\": true, \"question_groups\": [{\"name\": \"Group\", \"pickingStrategy\": \"all-ordered\", \"pickQuestions\": 1, \"questions\": [{\"name\": \"Add/Subtract 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/\"}, {\"name\": \"Shaheen Charlwood\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1819/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eLinear combinations of $2 \\\\times 2$ matrices. Three examples.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003eLet\u0026nbsp;\u003cbr/\u003e\\\\[A=\\\\simplify{{a}},\\\\;\\\\; B=\\\\simplify{{b}}\\\\;\\\\]\u003cbr/\u003eCalculate the following $2 \\\\times 2$ matrices:\u003c/p\u003e\\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\", \"advice\": \"\u003ch4\u003ea)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} \\\\simplify[std]{A+B} \u0026amp;=\u0026amp;\\\\simplify[std]{{a}+{b}}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[std]{{a[0][0]}+{b[0][0]}}\u0026amp;\u0026nbsp;\\\\simplify[std]{{a[0][1]}+{b[0][1]}}\\\\\\\\\u0026nbsp;\u0026nbsp;\\\\simplify[std]{{a[1][0]}+{b[1][0]}}\u0026amp;\\\\simplify[std]{{a[1][1]}+{b[1][1]}} \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp;\\\\simplify{{apb}}\\\\\\\\ \u0026nbsp;\\\\end{eqnarray*} \\\\]\u003c/p\u003e\\n\u003ch4\u003eb)\u0026nbsp;\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} \\\\simplify[std]{{p}A+{q}B} \u0026amp;=\u0026amp;\\\\simplify[std]{{p}{a}+{q}{b}}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[std]{{p}*{a[0][0]}+{q}*{b[0][0]}}\u0026amp;\u0026nbsp;\\\\simplify[std]{{p}*{a[0][1]}+{q}*{b[0][1]}}\\\\\\\\\u0026nbsp;\u0026nbsp;\\\\simplify[std]{{p}*{a[1][0]}+{q}*{b[1][0]}}\u0026amp;\\\\simplify[std]{{p}*{a[1][1]}+{q}*{b[1][1]}} \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp;\\\\simplify{{lcab}}\\\\\\\\ \u0026nbsp;\\\\end{eqnarray*} \\\\]\u003c/p\u003e\\n\u003ch4\u003ec)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} \\\\simplify[std]{{p1}A+{q1}B+{r1}C} \u0026amp;=\u0026amp;\\\\simplify[std]{{p1}{a}+{q1}{b}+{r1}{c}}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[std]{{p1}*{a[0][0]}+{q1}*{b[0][0]}+{r1}*{c[0][0]}}\u0026amp;\u0026nbsp;\\\\simplify[std]{{p1}*{a[0][1]}+{q1}*{b[0][1]}+{r1}*{c[0][1]}}\\\\\\\\\u0026nbsp;\u0026nbsp;\\\\simplify[std]{{p1}*{a[1][0]}+{q1}*{b[1][0]}+{r1}*{c[1][0]}}\u0026amp;\\\\simplify[std]{{p1}*{a[1][1]}+{q1}*{b[1][1]}+{r1}*{c[1][1]}} \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp;\\\\simplify{{lcabc}}\\\\\\\\ \u0026nbsp;\\\\end{eqnarray*} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\", \"rulesets\": {\"std\": [\"all\", \"fractionNumbers\", \"!collectNumbers\", \"!noleadingminus\"]}, \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"matrix(repeat(repeat(random(-5..5 except 0),2),2))\", \"description\": \"\", \"templateType\": \"anything\"}, \"q1\": {\"name\": \"q1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-6..6 except [0,1,-1,p1,q])\", \"description\": \"\", \"templateType\": \"anything\"}, \"c\": {\"name\": \"c\", \"group\": 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\"templateType\": \"anything\"}, \"apb\": {\"name\": \"apb\", \"group\": \"Ungrouped variables\", \"definition\": \"a+b\", \"description\": \"\", \"templateType\": \"anything\"}, \"lcab\": {\"name\": \"lcab\", \"group\": \"Ungrouped variables\", \"definition\": \"p*a+q*b\", \"description\": \"\", \"templateType\": \"anything\"}, \"lcabc\": {\"name\": \"lcabc\", \"group\": \"Ungrouped variables\", \"definition\": \"p1*a+q1*b+r1*c\", \"description\": \"\", \"templateType\": \"anything\"}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"q1\", \"c\", \"b\", \"r1\", \"q\", \"p\", \"p1\", \"apb\", \"lcab\", \"lcabc\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003e$\\\\mathrm{A}+\\\\mathrm{B} = \\\\simplify[std]{{a}+{b}} = $ [[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"apb\", \"correctAnswerFractions\": false, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": false, \"allowFractions\": true}], \"sortAnswers\": false}, {\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": false, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"$\\\\simplify{{p}A+{q}B = {p}{a}+{q}{b}}=$ [[0]]\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"showCorrectAnswer\": true, \"showFeedbackIcon\": false, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"lcab\", \"correctAnswerFractions\": false, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": false, \"allowFractions\": false}], \"sortAnswers\": false}], \"type\": \"question\"}, {\"name\": \"Matrix Multiplication\", \"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/\"}, {\"name\": \"Shaheen Charlwood\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1819/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eMultiplication of $2 \\\\times 2$ matrices.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\\n  \\n  \\n  \u003cp\u003eDo the following matrix problems \u003cbr/\u003eLet \u003cbr/\u003e\\\\[A=\\\\begin{pmatrix} \\\\var{a11}\u0026#38;\\\\var{a12}\\\\\\\\ \\\\var{a21}\u0026#38;\\\\var{a22}\\\\\\\\ \\\\end{pmatrix},\\\\;\\\\;\\n  \\n  B=\\\\begin{pmatrix} \\\\var{b11}\u0026#38;\\\\var{b12}\\\\\\\\ \\\\var{b21}\u0026#38;\\\\var{b22}\\\\\\\\ \\\\end{pmatrix},\\\\;\\\\;\\n  \\n  C=\\\\begin{pmatrix} \\\\var{c11}\u0026#38;\\\\var{c12}\\\\\\\\ \\\\var{c21}\u0026#38;\\\\var{c22}\\\\\\\\ \\\\end{pmatrix}\\\\]\u003cbr/\u003eCalculate the following products of these matrices:\u003c/p\u003e\\n  \\n  \\n    \\n    \", \"advice\": \"\u003ch4\u003ea)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} AB \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{a11}\u0026amp;\\\\var{a12}\\\\\\\\ \\\\var{a21}\u0026amp;\\\\var{a22}\\\\\\\\ \\\\end{pmatrix}\\\\begin{pmatrix} \\\\var{b11}\u0026amp;\\\\var{b12}\\\\\\\\ \\\\var{b21}\u0026amp;\\\\var{b22}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[]{{a11}{b11}+{a12}{b21}}\u0026amp;\\\\simplify[]{{a11}{b12}+{a12}{b22}}\\\\\\\\ \\\\simplify[]{{a21}{b11}+{a22}{b21}}\u0026amp;\\\\simplify[]{{a21}{b12}+{a22}{b22}}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{ab11}\u0026amp;\\\\var{ab12}\\\\\\\\ \\\\var{ab21}\u0026amp;\\\\var{ab22}\\\\\\\\ \\\\end{pmatrix} \\\\end{eqnarray*} \\\\]\u003c/p\u003e\\n\u003ch4\u003eb)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} BA \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{b11}\u0026amp;\\\\var{b12}\\\\\\\\ \\\\var{b21}\u0026amp;\\\\var{b22}\\\\\\\\ \\\\end{pmatrix}\\\\begin{pmatrix} \\\\var{a11}\u0026amp;\\\\var{a12}\\\\\\\\ \\\\var{a21}\u0026amp;\\\\var{a22}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[]{{b11}{a11}+{b12}{a21}}\u0026amp;\\\\simplify[]{{b11}{a12}+{b12}{a22}}\\\\\\\\ \\\\simplify[]{{b21}{a11}+{b22}{a21}}\u0026amp;\\\\simplify[]{{b21}{a12}+{b22}{a22}}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{ba11}\u0026amp;\\\\var{ba12}\\\\\\\\ \\\\var{ba21}\u0026amp;\\\\var{ba22}\\\\\\\\ \\\\end{pmatrix} \\\\end{eqnarray*} \\\\]\u003c/p\u003e\\n\u003ch4\u003ec)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} CB \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{c11}\u0026amp;\\\\var{c12}\\\\\\\\ \\\\var{c21}\u0026amp;\\\\var{c22}\\\\\\\\ \\\\end{pmatrix}\\\\begin{pmatrix} \\\\var{b11}\u0026amp;\\\\var{b12}\\\\\\\\ \\\\var{b21}\u0026amp;\\\\var{b22}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[]{{c11}{b11}+{c12}{b21}}\u0026amp;\\\\simplify[]{{c11}{b12}+{c12}{b22}}\\\\\\\\ \\\\simplify[]{{c21}{b11}+{c22}{b21}}\u0026amp;\\\\simplify[]{{c21}{b12}+{a22}{b22}}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{cb11}\u0026amp;\\\\var{cb12}\\\\\\\\ \\\\var{cb21}\u0026amp;\\\\var{cb22}\\\\\\\\ \\\\end{pmatrix} \\\\end{eqnarray*} \\\\]\u003c/p\u003e\\n\u003ch4\u003ed)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} AC \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{a11}\u0026amp;\\\\var{a12}\\\\\\\\ \\\\var{a21}\u0026amp;\\\\var{a22}\\\\\\\\ \\\\end{pmatrix}\\\\begin{pmatrix} \\\\var{c11}\u0026amp;\\\\var{c12}\\\\\\\\ \\\\var{c21}\u0026amp;\\\\var{c22}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[]{{a11}{c11}+{a12}{c21}}\u0026amp;\\\\simplify[]{{a11}{c12}+{a12}{c22}}\\\\\\\\ \\\\simplify[]{{a21}{c11}+{a22}{c21}}\u0026amp;\\\\simplify[]{{a21}{c12}+{a22}{c22}}\\\\\\\\ \\\\end{pmatrix}\\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\var{ac11}\u0026amp;\\\\var{ac12}\\\\\\\\ \\\\var{ac21}\u0026amp;\\\\var{ac22}\\\\\\\\ \\\\end{pmatrix} \\\\end{eqnarray*} \\\\]\u003c/p\u003e\", \"rulesets\": {\"std\": [\"all\", \"fractionNumbers\", \"!collectNumbers\"]}, \"variables\": {\"ba21\": {\"name\": \"ba21\", \"group\": \"Ungrouped variables\", \"definition\": \"b21*a11+b22*a21\", \"description\": \"\", \"templateType\": \"anything\"}, \"a21\": {\"name\": \"a21\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-2..2)\", \"description\": \"\", \"templateType\": \"anything\"}, \"a22\": {\"name\": \"a22\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..3)\", \"description\": \"\", \"templateType\": \"anything\"}, \"ba22\": {\"name\": \"ba22\", \"group\": \"Ungrouped variables\", \"definition\": \"b21*a12+b22*a22\", \"description\": \"\", \"templateType\": \"anything\"}, \"cb21\": {\"name\": \"cb21\", \"group\": \"Ungrouped variables\", \"definition\": \"c21*b11+c22*b21\", \"description\": \"\", \"templateType\": \"anything\"}, \"b22\": {\"name\": \"b22\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-3..-1)\", \"description\": \"\", \"templateType\": \"anything\"}, \"b21\": {\"name\": \"b21\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2,3)\", \"description\": \"\", \"templateType\": \"anything\"}, \"cb22\": {\"name\": \"cb22\", \"group\": \"Ungrouped variables\", \"definition\": \"c21*b12+c22*b22\", \"description\": \"\", \"templateType\": \"anything\"}, \"ac22\": {\"name\": \"ac22\", \"group\": \"Ungrouped variables\", \"definition\": 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variables\", \"definition\": \"a11*c11+a12*c21\", \"description\": \"\", \"templateType\": \"anything\"}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"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\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003e$AB = \\\\begin{pmatrix} \\\\var{a11}\u0026amp;\\\\var{a12}\\\\\\\\ \\\\var{a21}\u0026amp;\\\\var{a22}\\\\\\\\ \\\\end{pmatrix}\\\\begin{pmatrix} \\\\var{b11}\u0026amp;\\\\var{b12}\\\\\\\\ \\\\var{b21}\u0026amp;\\\\var{b22}\\\\\\\\ \\\\end{pmatrix} = $ [[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"matrix([\\n    [ab11,ab12],\\n    [ab21,ab22]\\n])\", \"correctAnswerFractions\": false, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": false, \"allowFractions\": false}], \"sortAnswers\": false}, {\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003e$CB = \\\\begin{pmatrix} \\\\var{c11}\u0026amp;\\\\var{c12}\\\\\\\\ \\\\var{c21}\u0026amp;\\\\var{c22}\\\\\\\\ \\\\end{pmatrix} \\\\begin{pmatrix} \\\\var{b11}\u0026amp;\\\\var{b12}\\\\\\\\ \\\\var{b21}\u0026amp;\\\\var{b22}\\\\\\\\ \\\\end{pmatrix}=$\u0026nbsp;[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"matrix([\\n    [cb11,cb12],\\n    [cb21,cb22]\\n])\", \"correctAnswerFractions\": false, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": false, \"allowFractions\": false}], \"sortAnswers\": false}], \"type\": \"question\"}, {\"name\": \"Shaheen\u0027s copy of Shaheen\u0027s copy of Priyanka\u0027s copy of Find determinants and inverses of 2x2 matrices\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Newcastle University Mathematics and Statistics\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/697/\"}, {\"name\": \"Shaheen Charlwood\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1819/\"}, {\"name\": \"Priyanka Pandya\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4134/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eFind the determinant and inverse of three $2 \\\\times 2$ invertible matrices.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003eLet\u003c/p\u003e\\n\u003cp\u003e\\\\begin{align} \\\\mathbf{A} \u0026amp;= \\\\var{a}, \u0026amp; \\\\mathbf{B} \u0026amp;= \\\\var{b}, \u0026amp; \\\\mathbf{C} \u0026amp;= \\\\var{c} \\\\end{align}\u003c/p\u003e\", \"advice\": \"\u003ch4\u003eDeterminant of a $2 \\\\times 2$ matrix\u003c/h4\u003e\\n\u003cp\u003eThe determinant of a matrix $\\\\mathrm{M} = \\\\begin{pmatrix} a\u0026amp;b \\\\\\\\ c\u0026amp;d \\\\end{pmatrix}$ is given by\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\det\\\\left(\\\\mathrm{M}\\\\right) = ad-bc \\\\]\u003c/p\u003e\\n\u003cp\u003eIf we have two $n \\\\times n$ matrices $M$ and $N$, then\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\det\\\\left(\\\\mathrm{MN}\\\\right) = \\\\det\\\\left(\\\\mathrm{M}\\\\right)\\\\det\\\\left(\\\\mathrm{N}\\\\right) \\\\]\u003c/p\u003e\\n\u003cp\u003eAnd it follows that if we have a third matrix $P$,\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\det\\\\left(\\\\mathrm{MNP}\\\\right) = \\\\det\\\\left(\\\\mathrm{M}\\\\right)\\\\det\\\\left(\\\\mathrm{N}\\\\right)\\\\det\\\\left(\\\\mathrm{P}\\\\right) \\\\]\u003c/p\u003e\\n\u003ch4\u003ea)\u003c/h4\u003e\\n\u003cp\u003eThus for our example we have:\u003c/p\u003e\\n\u003cp\u003e\\\\begin{align}\u003cbr/\u003e\\\\det\\\\left(\\\\mathrm{A}\\\\right) \u0026amp;= \\\\simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\\\\\\\\u003cbr/\u003e\\\\det\\\\left(\\\\mathrm{B}\\\\right) \u0026amp;= \\\\simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\\\\\\\\u003cbr/\u003e\\\\det\\\\left(\\\\mathrm{C}\\\\right) \u0026amp;= \\\\simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}}\u003cbr/\u003e\\\\end{align}\u003c/p\u003e\\n\u003cp\u003e\\\\begin{align}\u003cbr/\u003e\\\\det\\\\left( \\\\mathrm{ABC} \\\\right) \u0026amp;= \\\\det(\\\\mathrm{A}) \\\\det(\\\\mathrm{B}) \\\\det(\\\\mathrm{C}) \\\\\\\\\u003cbr/\u003e\u0026amp;= \\\\simplify[]{{det(a)}*{det(b)}*{det(c)}} \\\\\\\\\u003cbr/\u003e\u0026amp;= \\\\var{det(a*b*c)}\u003cbr/\u003e\\\\end{align}\u003c/p\u003e\\n\u003ch4\u003eInverse of a $2 \\\\times 2$ matrix\u003c/h4\u003e\\n\u003cp\u003eSuppose $\\\\mathrm{M} = \\\\begin{pmatrix} a\u0026amp;b \\\\\\\\ c\u0026amp;d \\\\end{pmatrix}$ is a $2 \\\\times 2$ matrix and $\\\\det\\\\left(\\\\mathrm{M}\\\\right) = \\\\Delta \\\\neq 0$.\u003c/p\u003e\\n\u003cp\u003eThen $\\\\mathrm{M}$ is invertible and\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\mathrm{M}^{-1} = \\\\frac{1}{\\\\Delta} \\\\begin{pmatrix} d \u0026amp; -b\\\\\\\\ -c\u0026amp; a \\\\end{pmatrix}=\\\\begin{pmatrix} \\\\frac{d}{\\\\Delta} \u0026amp; -\\\\frac{b}{\\\\Delta}\\\\\\\\ -\\\\frac{c}{\\\\Delta}\u0026amp; \\\\frac{a}{\\\\Delta} \\\\end{pmatrix}\\\\]\u003c/p\u003e\\n\u003cp\u003eApplying this to these examples we obtain:\u003c/p\u003e\\n\u003ch4\u003eb)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \\\\]\u003c/p\u003e\\n\u003ch4\u003ec)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \\\\]\u003c/p\u003e\\n\u003ch4\u003ed)\u003c/h4\u003e\\n\u003cp\u003e\\\\[ \\\\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \\\\]\u003c/p\u003e\", \"rulesets\": {\"std\": [\"all\", \"fractionNumbers\", \"!collectNumbers\"]}, \"variables\": {\"b22\": {\"name\": \"b22\", \"group\": \"Ungrouped variables\", \"definition\": \"if(tr3*b11=b21*b12,tr3+1,tr3)\", \"description\": \"\", \"templateType\": \"anything\"}, \"b21\": {\"name\": \"b21\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-6..6 except 0)\", \"description\": \"\", \"templateType\": \"anything\"}, \"tr2\": {\"name\": \"tr2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"b\": {\"name\": \"b\", \"group\": \"Unnamed group\", \"definition\": \"matrix([ [b11,b12], [b21,b22] ])\", \"description\": \"\", \"templateType\": \"anything\"}, \"c\": {\"name\": \"c\", \"group\": \"Unnamed group\", \"definition\": \"matrix([ [c11,c12], [c21,c22] ])\", \"description\": \"\", \"templateType\": \"anything\"}, \"a22\": {\"name\": \"a22\", \"group\": \"Ungrouped variables\", \"definition\": \"if(tr1*a11=a21*a12,tr1+1,tr1)\", \"description\": \"\", \"templateType\": \"anything\"}, \"c21\": {\"name\": \"c21\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..5)\", \"description\": \"\", \"templateType\": \"anything\"}, \"tr3\": {\"name\": \"tr3\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-9..9 except 0)\", \"description\": \"\", \"templateType\": \"anything\"}, \"c22\": {\"name\": \"c22\", \"group\": \"Ungrouped variables\", \"definition\": \"if(tr4*c11=c21*c12,tr4+1,tr4)\", \"description\": \"\", \"templateType\": \"anything\"}, \"c11\": {\"name\": \"c11\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,2,4)\", \"description\": \"\", \"templateType\": \"anything\"}, \"tr4\": {\"name\": \"tr4\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"b11\": {\"name\": \"b11\", \"group\": \"Ungrouped variables\", \"definition\": \"if(a11=tr2,tr2+1,tr2)\", \"description\": \"\", \"templateType\": \"anything\"}, \"tr1\": {\"name\": \"tr1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"a\": {\"name\": \"a\", \"group\": \"Unnamed group\", \"definition\": \"matrix([ [a11,a12],[a21,a22] ])\", \"description\": \"\", \"templateType\": \"anything\"}, \"a12\": {\"name\": \"a12\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5)\", \"description\": \"\", \"templateType\": \"anything\"}, \"b12\": {\"name\": \"b12\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5)\", \"description\": \"\", \"templateType\": \"anything\"}, \"c12\": {\"name\": \"c12\", \"group\": \"Ungrouped variables\", \"definition\": \"a12+b12\", \"description\": \"\", \"templateType\": \"anything\"}, \"a21\": {\"name\": \"a21\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-6..6 except 0)\", \"description\": \"\", \"templateType\": \"anything\"}, \"a11\": {\"name\": \"a11\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-9..9 except 0)\", \"description\": \"\", \"templateType\": \"anything\"}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a11\", \"a12\", \"a21\", \"a22\", \"b11\", \"b12\", \"b21\", \"b22\", \"c11\", \"c12\", \"c21\", \"c22\", \"tr1\", \"tr2\", \"tr3\", \"tr4\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": [\"a\", \"b\", \"c\"]}], \"functions\": {\"inverse\": {\"parameters\": [[\"m\", \"matrix\"]], \"type\": \"matrix\", \"language\": \"jme\", \"definition\": \"matrix([\\n    [m[1][1], -m[0][1]],\\n    [-m[1][0], m[0][0]]\\n])/det(m)\"}}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003eCalculate the determinants of these matrices.\u003c/p\u003e\\n\u003cp\u003e$\\\\mathrm{det}\\\\left(A\\\\right) = $\u0026nbsp;[[0]]\u003c/p\u003e\\n\u003cp\u003e$\\\\mathrm{det}\\\\left(B\\\\right) = $\u0026nbsp;[[1]]\u003c/p\u003e\\n\u003cp\u003e$\\\\mathrm{det}\\\\left(C\\\\right) = $\u0026nbsp;[[2]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0.5, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"minValue\": \"det(a)\", \"maxValue\": \"det(a)\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}, {\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0.5, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"minValue\": \"det(b)\", \"maxValue\": \"det(b)\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}, {\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0.5, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"minValue\": \"det(c)\", \"maxValue\": \"det(c)\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"sortAnswers\": false}, {\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003eFind the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals.\u003c/p\u003e\\n\u003cp\u003e$\\\\mathbf{A}^{-1} = $\u0026nbsp;[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"2\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"inverse(a)\", \"correctAnswerFractions\": true, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": false, \"allowFractions\": true}], \"sortAnswers\": false}, {\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003e$\\\\mathbf{B}^{-1} = $\u0026nbsp;[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"2\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"inverse(b)\", \"correctAnswerFractions\": true, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": false, \"allowFractions\": true}], \"sortAnswers\": false}], \"type\": \"question\"}, {\"name\": \"Shaheen\u0027s copy of Shaheen\u0027s copy of Andrew\u0027s copy of Matrices:  Inverse Matrix Method\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Andrew Dunbar\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/770/\"}, {\"name\": \"Shaheen Charlwood\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1819/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003ePutting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.\u0026nbsp;\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003eWrite the following equations as a matrix equation \u003cbr/\u003e\\\\[AX=b\\\\]for a matrix $A$ and column vectors $X$ and $b$\u003cbr/\u003e\\\\[ \\\\begin{eqnarray*} \\\\simplify[std]{{a}x+{b}y}\u0026amp;=\u0026amp;\\\\var{c}\\\\\\\\ \\\\simplify[std]{{a1}x+{b1}y}\u0026amp;=\u0026amp;\\\\var{c1} \\\\end{eqnarray*} \\\\]\u003c/p\u003e\", \"advice\": \"\u003ch4\u003ea)\u003c/h4\u003e\\n\u003cp\u003eThe equations can be written in the matrix form:\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\begin{pmatrix} \\\\var{a} \u0026amp; \\\\var{b}\\\\\\\\ \\\\var{a1}\u0026amp;\\\\var{b1} \\\\end{pmatrix} \\\\begin{pmatrix} x \\\\\\\\ y \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\var{c} \\\\\\\\ \\\\var{c1} \\\\end{pmatrix}\\\\]\u003c/p\u003e\\n\u003ch4\u003eb)\u003c/h4\u003e\\n\u003cp\u003eSince $\\\\mathrm{det}(A) = \\\\simplify[]{{a}*{b1}-{b}*{a1}={dA}} \\\\neq 0$, $A$ is invertible and\u003c/p\u003e\\n\u003cp\u003e\\\\[A^{-1} = \\\\begin{pmatrix} \\\\simplify[std]{{b1}/{dA}}\u0026amp;\\\\simplify[std]{{-b}/{dA}}\\\\\\\\\\\\simplify[std]{{-a1}/{dA}}\u0026amp;\\\\simplify[std]{{a}/{dA}} \\\\end{pmatrix}\\\\]\u003c/p\u003e\\n\u003ch4\u003ec)\u003c/h4\u003e\\n\u003cp\u003eWe have:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{eqnarray*} A^{-1}b \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[std]{{b1}/{dA}}\u0026amp;\\\\simplify[std]{{-b}/{dA}}\\\\\\\\\\\\simplify[std]{{-a1}/{dA}}\u0026amp;\\\\simplify[std]{{a}/{dA}} \\\\end{pmatrix}\\\\begin{pmatrix} \\\\var{c}\\\\\\\\\\\\var{c1}\\\\end{pmatrix} \\\\\\\\ \u0026amp;=\u0026amp; \\\\begin{pmatrix} \\\\simplify[std]{{c*b1-c1*b}/{dA}}\\\\\\\\\\\\simplify[std]{{c1*a-c*a1}/{dA}}\\\\end{pmatrix} \\\\end{eqnarray*} \\\\]\u003c/p\u003e\\n\u003ch4\u003ed)\u003c/h4\u003e\\n\u003cp\u003eNote that $AX = b \\\\Rightarrow X = A^{-1}b$ hence we can read the solution from the last part as this gives:\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\begin{pmatrix} x\\\\\\\\y \\\\end{pmatrix} = \\\\begin{pmatrix} \\\\simplify[std]{{c*b1-c1*b}/{dA}}\\\\\\\\ \\\\simplify[std]{{c1*a-c*a1}/{dA}}\\\\end{pmatrix}\\\\]\u003c/p\u003e\\n\u003cp\u003eHence \\\\[\\\\begin{eqnarray*} x\u0026amp;=\u0026amp; \\\\simplify[std]{{c*b1-c1*b}/{dA}}\\\\\\\\ y\u0026amp;=\u0026amp; \\\\simplify[std]{{c1*a-c*a1}/{dA}} \\\\end{eqnarray*} \\\\]\u003c/p\u003e\", \"rulesets\": {\"std\": [\"all\", \"!collectNumbers\", \"fractionNumbers\", \"!noLeadingMinus\"]}, \"variables\": {\"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"sc*random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"sc1\": {\"name\": \"sc1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,-1)\", \"description\": \"\", \"templateType\": \"anything\"}, \"sa1\": {\"name\": \"sa1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,-1)\", \"description\": \"\", \"templateType\": \"anything\"}, \"sb1\": {\"name\": \"sb1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,-1)\", \"description\": \"\", \"templateType\": \"anything\"}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"sb*random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"sb\": {\"name\": \"sb\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,-1)\", \"description\": \"\", \"templateType\": \"anything\"}, \"b1\": {\"name\": \"b1\", \"group\": \"Ungrouped variables\", \"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)))))\", \"description\": \"\", \"templateType\": \"anything\"}, \"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"sa*random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"test\": {\"name\": \"test\", \"group\": \"Ungrouped variables\", \"definition\": \"round(a*b1/b)\", \"description\": \"\", \"templateType\": \"anything\"}, \"sa\": {\"name\": \"sa\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,-1)\", \"description\": \"\", \"templateType\": \"anything\"}, \"ab\": {\"name\": \"ab\", \"group\": \"Ungrouped variables\", \"definition\": \"abs(b)\", \"description\": \"\", \"templateType\": \"anything\"}, \"inc\": {\"name\": \"inc\", \"group\": \"Ungrouped variables\", \"definition\": \"sa*random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"a1\": {\"name\": \"a1\", \"group\": \"Ungrouped variables\", \"definition\": \"if(test+inc=0,test+inc+1,test+inc)\", \"description\": \"\", \"templateType\": \"anything\"}, \"c1\": {\"name\": \"c1\", \"group\": \"Ungrouped variables\", \"definition\": \"sc1*random(1..9)\", \"description\": \"\", \"templateType\": \"anything\"}, \"addortakeaway\": {\"name\": \"addortakeaway\", \"group\": \"Ungrouped variables\", \"definition\": \"if(b*b1\u003c0,\u0027add the equation\u0027,\u0027take away the equation\u0027)\", \"description\": \"\", \"templateType\": \"anything\"}, \"da\": {\"name\": \"da\", \"group\": \"Ungrouped variables\", \"definition\": \"a*b1-a1*b\", \"description\": \"\", \"templateType\": \"anything\"}, \"sc\": {\"name\": \"sc\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,-1)\", \"description\": \"\", \"templateType\": \"anything\"}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"addortakeaway\", \"c\", \"b\", \"da\", \"a1\", \"sc\", \"ab\", \"sb1\", \"b1\", \"test\", \"sb\", \"sa\", \"sc1\", \"sa1\", \"c1\", \"inc\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp style=\\\"text-align: center;\\\"\u003e$A = $\u0026nbsp;[[0]]\u003c/p\u003e\\n\u003ctable style=\\\"margin-left: auto; margin-right: auto;\\\"\u003e\\n\u003ctbody\u003e\\n\u003ctr\u003e\\n\u003ctd rowspan=\\\"2\\\"\u003e$X = \\\\;\\\\;\\\\Bigg($\u003c/td\u003e\\n\u003ctd\u003e[[1]]\u003c/td\u003e\\n\u003ctd rowspan=\\\"2\\\"\u003e$\\\\Bigg)$\u003c/td\u003e\\n\u003c/tr\u003e\\n\u003ctr\u003e\\n\u003ctd\u003e[[2]]\u003c/td\u003e\\n\u003c/tr\u003e\\n\u003c/tbody\u003e\\n\u003c/table\u003e\\n\u003cp style=\\\"text-align: center;\\\"\u003e$b = $\u0026nbsp;[[3]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"matrix([\\n    [a,b],\\n    [a1,b1]\\n])\", \"correctAnswerFractions\": false, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": true}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"0.25\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"answer\": \"x\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": true, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"0.25\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"answer\": \"y\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": true, \"valuegenerators\": [{\"name\": \"y\", \"value\": \"\"}]}, {\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"0.5\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"matrix([\\n    [c],\\n    [c1]\\n])\", \"correctAnswerFractions\": true, \"numRows\": \"2\", \"numColumns\": 1, \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": true}], \"sortAnswers\": false}, {\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003e\u003cspan\u003eFind the inverse of $A$. \u003c/span\u003e\u003c/p\u003e\\n\u003cul\u003e\\n\u003cli\u003e\u003cspan\u003eInput all numbers as fractions or integers (not decimals). \u003c/span\u003e\u003c/li\u003e\\n\u003cli\u003e\u003cspan\u003eYou do not need to simplify any fractions.\u003c/span\u003e\u003c/li\u003e\\n\u003c/ul\u003e\\n\u003cp\u003e$A^{-1} = $\u0026nbsp;[[1]][[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"2\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"matrix([\\n    [b1,-b],\\n    [-a1,a]\\n])\", \"correctAnswerFractions\": true, \"numRows\": \"2\", \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": true}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"answer\": \"1/{da}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"valuegenerators\": []}], \"sortAnswers\": false}, {\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003eCalculate the following:\u003c/p\u003e\\n\u003cul\u003e\\n\u003cli\u003eInput your answers as integers or fractions (not decimals).\u003c/li\u003e\\n\u003cli\u003eYou do not need to simplify fractions.\u003c/li\u003e\\n\u003c/ul\u003e\\n\u003cp\u003e$A^{-1}b = $\u0026nbsp;[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"correctAnswer\": \"matrix([{c*b1-c1*b}/{b1*a-a1*b}],[{-c*a1+c1*a}/{b1*a-a1*b}])\", \"correctAnswerFractions\": true, \"numRows\": \"2\", \"numColumns\": 1, \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": true}], \"sortAnswers\": false}, {\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"prompt\": \"\u003cp\u003eNow solve the equations, inputting your answers as fractions or integers (not decimals).\u003c/p\u003e\\n\u003cp\u003e\u003cbr/\u003e$x = \\\\;\\\\;$[[0]]\u003c/p\u003e\\n\u003cp\u003e$y = \\\\;\\\\;$[[1]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"0.25\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"answer\": \"{c*b1-c1*b}/{b1*a-a1*b}\", \"answerSimplification\": \"std\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"notallowed\": {\"strings\": [\".\"], \"showStrings\": false, \"partialCredit\": 0, \"message\": \"\u003cp\u003eInput as a fraction or an integer, not as a decimal\u003c/p\u003e\"}, \"valuegenerators\": []}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"0.25\", \"showCorrectAnswer\": true, \"showFeedbackIcon\": false, \"scripts\": {}, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"adaptiveMarkingPenalty\": 0, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"answer\": \"{-c*a1+c1*a}/{b1*a-a1*b}\", \"answerSimplification\": \"std\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"notallowed\": {\"strings\": [\".\"], \"showStrings\": false, \"partialCredit\": 0, \"message\": \"\u003cp\u003eInput as a fraction or an integer, not as a decimal\u003c/p\u003e\"}, \"valuegenerators\": []}], \"sortAnswers\": false}], \"type\": \"question\"}, {\"name\": \"Shaheen\u0027s copy of Cramers Rule 3x3\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [\"question-resources/Cramers.PNG\"], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Shaheen Charlwood\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1819/\"}, {\"name\": \"Marisa Llorens\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/2155/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCramers Rule applied to 3 simultaneous equations\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003e\u003cimg width=\\\"876\\\" height=\\\"362\\\" src=\\\"resources/question-resources/Cramers.PNG\\\"/\u003e\u003c/p\u003e\\n\u003cp\u003eUsing Cramer\u0027s rule (above), solve these simultaneous equations:\u003c/p\u003e\\n\u003cp\u003e$\\\\var{a11}x+\\\\var{a12}y+\\\\var{a13}z=\\\\var{c1}$\u003c/p\u003e\\n\u003cp\u003e$\\\\var{a21}x+\\\\var{a22}y+\\\\var{a23}z=\\\\var{c2}$\u003c/p\u003e\\n\u003cp\u003e$\\\\var{a31}x+\\\\var{a32}y+\\\\var{a33}z=\\\\var{c3}$\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIf \\\\[\u0026nbsp; A=\\\\left( \\\\begin{array}{ccc}\u003cbr/\u003ea_{11} \u0026amp;\u0026nbsp;a_{12} \u0026amp; a_{13} \\\\\\\\a_{21} \u0026amp;\u0026nbsp;a_{22} \u0026amp; a_{23}\\\\\\\\ a_{31} \u0026amp;\u0026nbsp;a_{32} \u0026amp; a_{33}\\\\end{array} \\\\right),\\\\]\u003c/p\u003e\\n\u003cp\u003e\\\\[\u0026nbsp; C=\\\\left( \\\\begin{array}{ccc}\u003cbr/\u003ec_{1} \\\\\\\\\u0026nbsp;c_{2}\u0026nbsp;\\\\\\\\c_{3} \\\\end{array} \\\\right),\\\\]\u003c/p\u003e\\n\u003cp\u003eCramer\u0027s Rule : ${x_1}=\\\\frac{\\\\Delta_1}{\\\\Delta_0}$ ,\u0026nbsp; ${x_2}=\\\\frac{\\\\Delta_2}{\\\\Delta_0}$ , ${x_3}=\\\\frac{\\\\Delta_3}{\\\\Delta_0}$\u003c/p\u003e\\n\u003cp\u003eWhere:\\\\[ \\\\Delta_0=\\\\left| \\\\begin{array}{ccc}\u003cbr/\u003ea_{11} \u0026amp;\u0026nbsp;a_{12} \u0026amp; a_{13} \\\\\\\\a_{21} \u0026amp;\u0026nbsp;a_{22} \u0026amp; a_{23}\\\\\\\\ a_{31} \u0026amp;\u0026nbsp;a_{32} \u0026amp; a_{33}\\\\end{array} \\\\right|\\\\]\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\Delta_1=\\\\left| \\\\begin{array}{ccc}\u003cbr/\u003ec_{1} \u0026amp;\u0026nbsp;a_{12} \u0026amp; a_{13} \\\\\\\\c_{2} \u0026amp;\u0026nbsp;a_{22} \u0026amp; a_{23}\\\\\\\\ c_{3} \u0026amp;\u0026nbsp;a_{32} \u0026amp; a_{33}\\\\end{array} \\\\right|\\\\]\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\Delta_2=\\\\left| \\\\begin{array}{ccc}\u003cbr/\u003ea_{11} \u0026amp;\u0026nbsp;c_{1} \u0026amp; a_{13} \\\\\\\\a_{21} \u0026amp;\u0026nbsp;c_{2} \u0026amp; a_{23}\\\\\\\\ a_{31} \u0026amp;\u0026nbsp;c_{3} \u0026amp; a_{33}\\\\end{array} \\\\right|\\\\]\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\Delta_3=\\\\left| \\\\begin{array}{ccc}\u003cbr/\u003ea_{11} \u0026amp;\u0026nbsp;a_{12} \u0026amp; c_{1} \\\\\\\\a_{21} \u0026amp;\u0026nbsp;a_{22} \u0026amp; c_{2}\\\\\\\\ a_{31} \u0026amp;\u0026nbsp;a_{32} \u0026amp; c_{3}\\\\end{array} \\\\right|\\\\]\u003c/p\u003e\\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003e\u003cspan class=\\\"nb\\\"\u003e\u003cspan class=\\\"k\\\"\u003e\u003cspan class=\\\"s\\\"\u003e\u0026nbsp;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"rulesets\": {}, \"variables\": {\"matrixA2\": {\"name\": \"matrixA2\", \"group\": \"Cramer determinants\", \"definition\": \"matrix([a11,c1,a13],[a21,c2,a23],[a31,c3,a33])\", \"description\": \"\", \"templateType\": \"anything\"}, \"a33\": {\"name\": \"a33\", \"group\": \"Ungrouped variables\", \"definition\": \"random(0..10)\", \"description\": \"\", \"templateType\": \"anything\"}, \"a22\": {\"name\": \"a22\", \"group\": \"Ungrouped variables\", \"definition\": \"random(0..10 except(a21*a12/a11))\", \"description\": \"\", \"templateType\": \"anything\"}, \"matrixA3\": {\"name\": \"matrixA3\", \"group\": \"Cramer determinants\", \"definition\": \"matrix([a11,a12,c1],[a21,a22,c2],[a31,a32,c3])\", \"description\": \"\", \"templateType\": \"anything\"}, \"matrixA1\": {\"name\": \"matrixA1\", \"group\": \"Cramer 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\"oncompletion\", \"onleave\": {\"action\": \"warnifunattempted\", \"message\": \"\u003cp\u003eYou have not answered this question!\u003c/p\u003e\"}, \"preventleave\": true, \"startpassword\": \"y1aii\"}, \"timing\": {\"allowPause\": false, \"timeout\": {\"action\": \"warn\", \"message\": \"\u003cp\u003eThe matrices part of your in-class assessment is now finished .\u003c/p\u003e\"}, \"timedwarning\": {\"action\": \"warn\", \"message\": \"\u003cp\u003eYou have 5 minutes left to completes your matrices assessment.\u003c/p\u003e\"}}, \"feedback\": {\"showactualmark\": false, \"showtotalmark\": true, \"showanswerstate\": false, \"allowrevealanswer\": false, \"advicethreshold\": 0, \"intro\": \"\u003cp\u003eThis part of the in-class assessment\u003c/p\u003e\\n\u003cul\u003e\\n\u003cli\u003eassesses matrices: matrix arithmetic, determinants and inverses, solving simultaneous equations using matrix methods\u0026nbsp;\u003c/li\u003e\\n\u003cli\u003eis worth 20% of your overall WM104 module mark.\u003c/li\u003e\\n\u003c/ul\u003e\\n\u003cp\u003eAnswer all of the questions.\u003c/p\u003e\\n\u003cp\u003eIf you need anything during the test please raise your hand to alert the invigilator.\u003c/p\u003e\\n\u003cp\u003eYou may use rough paper for your calculations before inputting your answers.\u003c/p\u003e\\n\u003cp\u003eRemember tio use the correct format (as stated in the question) for your answer.\u003c/p\u003e\", \"feedbackmessages\": [], \"enterreviewmodeimmediately\": true, \"showexpectedanswerswhen\": \"inreview\", \"showpartfeedbackmessageswhen\": \"oncompletion\", \"showactualmarkwhen\": \"oncompletion\", \"showtotalmarkwhen\": \"always\", \"showanswerstatewhen\": \"oncompletion\", \"showadvicewhen\": \"never\"}, \"type\": \"exam\", \"contributors\": [{\"name\": \"Marie Nicholson\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1799/\"}, {\"name\": \"Shaheen Charlwood\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1819/\"}], \"extensions\": [\"stats\"], \"custom_part_types\": [], \"resources\": [[\"question-resources/Cramers.PNG\", \"/srv/numbas/media/question-resources/Cramers.PNG\"]]}",
            extension_data: "{\"stats\": {\"root\": \"extensions/stats\", \"stylesheets\": [], \"javascripts\": [\"stats.js\"]}}"
        }
    }
</script>



	</head>
	
    <body>
        <template id="numbas-exam-template">
    
    <style id="numbas-style">@import 'numbas.css?build_time=1764939318.806644';</style>
    
    <exam-container class="numbas-container" data-bind="with: exam, css: css, attr: attr">
        <div id="loading">
    <img class="logo center-block" src="resources/numbas-logo.svg" alt="Numbas"/>
    <progress min="0" max="100" value="0"></progress>
    <span class="loading-text" data-localise="page.loading">Loading...</span>
</div>
        <!-- modal dialog boxes -->
<dialog id="saving" data-bind="modal: saving">
    <div class="text" data-localise="page.saving"></div>
</dialog>
<dialog class="modal" id="alert-modal" aria-labelledby="alert-modal-title" aria-describedby="alert-modal-body" data-bind="modal: {submit: $root.modal.ok}">
    <section>
        <header>
            <h2 id="alert-modal-title" data-localise="modal.alert"></h2>
        </header>

        <div class="modal-body" id="alert-modal-body"></div>

        <footer>
            <form method="dialog">
                <button value="ok" autofocus class="ok btn primary" data-localise="modal.ok"></button>
            </form>
        </footer>
    </section>
</dialog>

<dialog class="modal" id="confirm-modal" aria-labelledby="confirm-modal-title" aria-describedby="confirm-modal-body" data-bind="modal: {submit: $root.modal.ok, cancel: $root.modal.cancel}">
    <section>
        <header>
            <h2 id="confirm-modal-title" data-localise="modal.confirm"></h2>
        </header>
        <div class="modal-body" id="confirm-modal-body">
        </div>
        <footer>
            <form method="dialog">
                <button autofocus class="cancel btn" value="cancel" data-localise="modal.cancel"></button>
                <button value="ok" class="ok btn primary" data-localise="modal.ok"></button>
            </form>
        </footer>
    </section>
</dialog>
<dialog class="modal" id="confirm-end-exam-modal" aria-labelledby="confirm-end-exam-modal-title" aria-describedby="confirm-end-exam-modal-body" data-bind="modal: {submit: $root.modal.ok, cancel: $root.modal.cancel}">
    <form method="dialog">
        <header>
            <h4 id="confirm-end-exam-modal-title" data-localise="modal.confirm"></h4>
        </header>
        <div class="modal-body" id="confirm-end-exam-modal-body" data-bind="with: $root.exam">
            <p id="confirm-end-exam-modal-message"></p>
            <form class="form-inline" data-bind="submit: $root.modal.ok">
                <label for="confirm-end-text-input" id="confirm-end-exam-modal-input-message"></label>
                <input autofocus id="confirm-end-text-input" type="text" data-bind="textInput: confirmEndHandler.value, autosize: confirmEndHandler.value">
                <span class="password-feedback feedback-icon" data-bind="css: confirmEndHandler.feedback().iconClass, attr: {title: confirmEndHandler.feedback().title}"></span>
            </form>
        </div>
        <footer>
            <button value="cancel" class="cancel btn" data-localise="modal.cancel"></button>
            <button value="ok" class="ok btn primary" data-bind="disable: !confirmEndHandler.valid(), css: confirmEndHandler.feedback().buttonClass" data-localise="modal.end exam button"></button>
        </footer>
    </form>
</dialog>
<dialog class="modal force-default-style" id="style-modal" aria-labelledby="style-modal-title" aria-describedby="style-modal-explanation" data-bind="with: $root.style_customiser, modal: {submit: $root.saveStyle}">
    <form method="dialog">
        <header>
            <h4 id="style-modal-title" data-localise="control.style options"></h4>
        </header>

        
        <p class="help-block" id="style-modal-explanation" data-localise="modal.style.explanation"></p>

        <p class="help-block warning" data-bind="visible: forced_colors" data-localise="modal.forced colours warning"></p>

        
        <section class="style-controls">
            <fieldset>
                <label for="main-font-picker" data-localise="modal.style.main font"></label>
                <select id="main-font-picker" data-bind="value: staged_style['--main-font'], options: font_options, optionsText: 'label', optionsValue: 'name'">
                </select>
            </fieldset>
            <fieldset>
                <label for="text-size-picker" data-localise="modal.style.text size"></label>
                <input id="text-size-picker" type="number" step="0.1" min="0.5" max="4" data-bind="value: staged_style['--text-size']">
                <output id="text-size-preview" class="big" data-localise="modal.style.text preview"></output>
            </fieldset>
            <fieldset>
                <label for="font-weight-picker" data-localise="modal.style.font weight"></label>
                <input id="font-weight-picker" type="range" step="100" min="200" max="800" data-bind="value: staged_style['--font-weight'], valueUpdate: 'input'">
                <output id="font-weight-preview" data-bind="text: staged_style['--font-weight']"></output>
            </fieldset>
            <fieldset>
                <label for="spacing-picker" data-localise="modal.style.spacing"></label>
                <input id="spacing-picker" type="range" step="0.1" min="0.1" max="3" data-bind="value: staged_style['--spacing-scale'], valueUpdate: 'input'">
                <output id="spacing-preview" data-bind="percentage: parseFloat(staged_style['--spacing-scale']())"></output>
            </fieldset>
            <fieldset>
                <label for="color-scheme-picker" data-localise="modal.style.colour scheme"></label>
                <select id="color-scheme-picker" data-bind="value: color_scheme">
                    <option value="automatic" data-localise="modal.style.automatic colour scheme"></option>
                    <option value="light" data-localise="modal.style.light colour scheme"></option>
                    <option value="dark" data-localise="modal.style.dark colour scheme"></option>
                    <option value="custom" data-localise="modal.style.custom colour scheme"></option>
                </select>
            </fieldset>
            <fieldset>
                <label for="invert-images" data-localise="modal.style.invert images in dark mode"></label>
                <input id="invert-images" type="checkbox" data-bind="checked: invert_images">
            </fieldset>
        </section>

        <section class="style-controls" data-bind="visible: using_custom_color_scheme">
            <fieldset>
                <label for="background-colour-picker" data-localise="modal.style.background colour"></label>
                <input id="background-colour-picker" type="color" data-bind="value: style['--custom-background-color']">
            </fieldset>
            <fieldset>
                <label for="text-colour-picker" data-localise="modal.style.text colour"></label>
                <input id="text-colour-picker" type="color" data-bind="value: style['--custom-text-color']">
            </fieldset>
            <fieldset>
                <label for="main-colour-picker" data-localise="modal.style.main colour"></label>
                <input id="main-colour-picker" type="color" data-bind="value: style['--custom-main-color']">
            </fieldset>
            <fieldset>
                <label for="primary-colour-picker" data-localise="modal.style.primary colour"></label>
                <input id="primary-colour-picker" type="color" data-bind="value: style['--custom-primary-color']">
            </fieldset>
            <fieldset>
                <label for="link-colour-picker" data-localise="modal.style.link colour"></label>
                <input id="link-colour-picker" type="color" data-bind="value: style['--custom-link-color']">
            </fieldset>
            <fieldset>
                <label for="success-colour-picker" data-localise="modal.style.success colour"></label>
                <input id="success-colour-picker" type="color" data-bind="value: style['--custom-success-color']">
            </fieldset>
            <fieldset>
                <label for="info-colour-picker" data-localise="modal.style.info colour"></label>
                <input id="info-colour-picker" type="color" data-bind="value: style['--custom-info-color']">
            </fieldset>
            <fieldset>
                <label for="warning-colour-picker" data-localise="modal.style.warning colour"></label>
                <input id="warning-colour-picker" type="color" data-bind="value: style['--custom-warning-color']">
            </fieldset>
            <fieldset>
                <label for="danger-colour-picker" data-localise="modal.style.danger colour"></label>
                <input id="danger-colour-picker" type="color" data-bind="value: style['--custom-danger-color']">
            </fieldset>
            <fieldset>
                <label for="muted-colour-picker" data-localise="modal.style.muted colour"></label>
                <input id="muted-colour-picker" type="color" data-bind="value: style['--custom-muted-color']">
            </fieldset>
            <fieldset>
                <label for="highlight-colour-picker" data-localise="modal.style.highlight colour"></label>
                <input id="highlight-colour-picker" type="color" data-bind="value: style['--custom-highlight-color']">
            </fieldset>
        </section>

        

        

        
        <hr/>
        <p class="help-block"><a href="https://docs.numbas.org.uk/en/latest/accessibility/exam.html" target="_blank" data-localise="accessibility statement"></a></p>
        

        <footer>
            <button type="button" class="btn danger sm" data-localise="modal.style.reset to defaults" data-bind="click: reset"></button>
            <button class="cancel btn" value="cancel" autofocus data-localise="modal.cancel"></button>
            <button value="ok" class="ok btn primary" data-localise="modal.ok"></button>
        </footer>
    </form>
</dialog>
<dialog class="modal" id="next-actions-modal" aria-labelledby="next-actions-modal-title">
    <form method="dialog" data-bind="with: diagnostic_next_actions">
        <header>
            <h4 id="next-actions-modal-title" data-localise="diagnostic.make a choice"></h4>
        </header>
        <div data-bind="latex: feedback"></div>

        <menu data-bind="foreach: actions" class="list-unstyled actions">
            <li>
                <button type="button" data-bind="click: go"><span data-bind="latex: label"></span></button>
            </li>
        </menu>
    </form>
</dialog>

<dialog id="lightbox"></dialog>
        <div id="everything">
            <a class="sr-only" id="skip-link" data-bind="attr: {href: infoPage() ? '#infoDisplay' : '#questionContainer'}" data-localise="page.skip to content"></a>
            <!-- ko if: exam.settings.navigateMode=='sequence' -->
    <!-- side menu, slides out on narrow screens -->
<nav id="sideNav">

    <header>
        <img class="logo center-block" src="resources/numbas-logo.svg" alt="Numbas"/>

        <h1 class="exam-name" data-bind="text: exam.settings.name, typeset: exam.settings.name"></h1>

        <button type="button" class="navbar-toggle btn" aria-controls="navMenu">
            <svg viewBox="0 0 9 9">
    <title data-localise="control.toggle navigation menu"></title>
    <line x1="1" y1="1" x2="8" y2="8" stroke="currentColor"/>
    <line x1="1" y1="8" x2="8" y2="1" stroke="currentColor"/>
</svg>
        </button>
    </header>

    <!-- ko if: numQuestions() > 1 -->
    <label class="sr-only" for="exam-progress" data-localise="exam.progress"></label>
    <progress id="exam-progress" data-bind="attr: {max: numQuestions(), value: questionsAttempted(), title: questionsAttemptedDisplay}, text: questionsAttemptedDisplay"></progress>
    <!-- /ko -->

    <menu id="questionList" class="nav navmenu-nav" role="tablist" aria-orientation="vertical" data-bind="tablist: true">
        <li class="introduction" data-bind="visible: exam.hasIntro, css: {active: $root.exam().infoPage()=='introduction'}">
            <a href="#" role="tab" data-bind="click: () => showInfoPage('introduction'), attr: {'aria-current': $root.exam().infoPage()=='introduction' ? 'step' : false, tabindex: $root.exam().infoPage()=='introduction' ? 0 : -1, 'data-name': R('control.show introduction')}" data-localise="control.show introduction"></a>
       </li>
        <!-- ko foreach: question_groups -->
            <!-- ko if: $root.exam().showQuestionGroupNames -->
            <li class="question-group-header" data-bind="visible: show_name">
                <h2 data-bind="text: name, typeset: name"></h2>
                <span class="score" data-bind="html: feedback.message"></span>
            </li>
            <!-- /ko -->
            <!-- ko foreach: questions -->
            <li class="questionSelector" data-bind="visible: visible, css: {active: $root.exam().currentQuestion()==$data}">
                <a href="#" role="tab" data-bind="click: Numbas.controls.makeQuestionJumper(question.number, question.exam), attr: {'aria-current': isCurrentQuestion() ? 'step' : false, tabindex: isCurrentQuestion() ? 0 : -1, 'data-number': question.number+1, 'data-name': displayName}">
                    <div class="question-info">
                        <span class="feedback-icon" data-bind="css: scoreFeedback.iconClass, attr: scoreFeedback.iconAttr" aria-hidden="true"></span>
                        <span class="name" data-bind="text: displayName"></span>
                        <span class="sr-only" data-bind="text: scoreFeedback.iconAttr().title"></span>
                        <span class="score" data-bind="html: scoreFeedback.message"></span>
                    </div>
                    <div class="messages">
                        <div class="answered-state" data-bind="">
                            <span data-bind="text: scoreFeedback.answeredString"></span>
                        </div>
                        <div data-bind="visible: isDirty" class="dirty-message">
                            <span class="warning-icon icon-exclamation-sign"></span>
                            <span data-localise="question.selector.unsubmitted changes"></span>
                        </div>
                    </div>
                </a>
            </li>
            <!-- /ko -->
        <!-- /ko -->
        <!-- total exam score display -->
        <li class="examScore" data-bind="visible: showActualMark">
            <span data-localise="control.total"></span> 
            <span id="examScore" data-bind="html: examScoreDisplay"></span>
        </li>
    </menu>

    <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>

    <menu class="nav navmenu-nav menuButtons">
        <li><button class="btn center-block thin-button display-options force-original-color" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button></li>
        <li data-bind="visible: allowPause && mode()=='normal'"><button class="btn center-block thin-button" id="pauseBtn" data-localise="control.pause" data-bind="click: pause"></button></li>
        <li data-bind="visible: mode()=='normal'"><button class="btn danger center-block" id="endBtn" data-localise="control.end exam" data-bind="click: endExam"></button></li>
        <li data-bind="visible: mode()=='review'"><button class="btn center-block" data-localise="control.back to results" data-bind="click: () => showInfoPage('result')"></button></li>
    </menu>

</nav>

<!-- top menu shown on narrow screens -->
<nav id="topNav" class="navbar">
    <!-- ko if: numQuestions() > 1 -->
    <div class="prev"><button type="button" class="btn" data-bind="click: reverse, attr: {disabled: !canReverse()}"><span class="sr-only" data-localise="control.previous"></span>&larr;</button></div>
    <div class="next"><button type="button" class="btn" data-bind="click: advance, attr: {disabled: !canAdvance()}"><span class="sr-only" data-localise="control.next"></span>&rarr;</span></button></div>
    <!-- /ko -->
    <div class="end"><button class="btn danger" data-localise="control.end exam" data-bind="click: endExam, visible: mode()=='normal' && !canAdvance()"></button></div>
    <div class="titleBits" data-bind="with: currentQuestion">
        <span class="review-header" data-bind="visible: $root.exam().mode()=='review'" data-localise="exam.review header"></span>
        <span id="questionNameDisplay" data-bind="text: displayName"></span>
    </div>

    <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>

    <button type="button" class="navbar-toggle btn" aria-controls="navMenu">
        <svg viewBox="0 0 9 9">
    <title data-localise="control.toggle navigation menu"></title>
    <rect x="0" y="0" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="4" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="8" width="9" height="1" fill="currentColor"/>
</svg>
    </button>
</nav>
<!-- /ko -->

<!-- ko if: exam.settings.navigateMode=='menu' -->
    <nav id="topNav" class="navbar">
    <button type="button" class="btn primary" data-bind="visible: mode()=='normal', click: showMenu">&larr; <span data-localise="control.back to menu"></span></button>
    <button type="button" class="btn primary" data-bind="visible: mode()=='review', click: () => showInfoPage('result')">&larr; <span data-localise="control.back to results"></span></button>
    <div class="titleBits" data-bind="with: currentQuestion">
        <span class="review-header" data-bind="visible: $root.exam().mode()=='review'" data-localise="exam.review header"></span>
        <span id="questionNameDisplay" data-bind="latex: displayName"></span>
    </div>
    <button class="btn thin-button display-options force-original-color" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button>
</nav>
<!-- /ko -->

<!-- ko if: exam.settings.navigateMode=='diagnostic' -->
    <!-- side menu, slides out on narrow screens -->
<nav id="sideNav">

    <header>
        <div class="navmenu-brand">
            <img class="logo center-block" src="resources/numbas-logo.svg" alt="Numbas"/>
        </div>
        <h1 class="exam-name navmenu-brand" data-bind="text: exam.settings.name, typeset: exam.settings.name"></h1>
        <button type="button" class="navbar-toggle btn" aria-controls="navMenu">
            <svg viewBox="0 0 9 9">
    <title data-localise="control.toggle navigation menu"></title>
    <line x1="1" y1="1" x2="8" y2="8" stroke="currentColor"/>
    <line x1="1" y1="8" x2="8" y2="1" stroke="currentColor"/>
</svg>
        </button>
    </header>

    <dl id="learning-objective-list" class="nav navemnu-nav">
        <!-- ko foreach: diagnostic_progress -->
        <dt class="name" data-bind="latex: name"></dt>
        <dd data-bind="text: Math.floor(100*progress)+'%'"></dd>
        <!-- /ko -->
    </dl>


    <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>

    <menu class="nav navmenu-nav menuButtons">
        <li><button class="btn center-block thin-button display-options force-original-color" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button></li>
        <li data-bind="visible: allowPause && mode()=='normal'"><button class="btn center-block thin-button" id="pauseBtn" data-localise="control.pause" data-bind="click: pause"></button></li>
        <li data-bind="visible: mode()=='normal'"><button class="btn danger center-block" id="endBtn" data-localise="control.end exam" data-bind="click: endExam"></button></li>
        <li data-bind="visible: mode()=='review'"><button class="btn center-block" data-localise="control.back to results" data-bind="click: () => showInfoPage('result')"></button></li>
    </menu>

</nav>

<!-- top menu shown on narrow screens -->
<nav id="topNav" class="navbar">
    <div class="nav">
        <div class="titleBits" data-bind="with: currentQuestion">
            <span class="review-header" data-bind="visible: $root.exam().mode()=='review'" data-localise="exam.review header"></span>
            <span id="questionNameDisplay" data-bind="text: displayName"></span>
        </div>

        <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>
    </div>

    <button type="button" class="navbar-toggle btn" aria-controls="navMenu">
        <svg viewBox="0 0 9 9">
    <title data-localise="control.toggle navigation menu"></title>
    <rect x="0" y="0" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="4" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="8" width="9" height="1" fill="currentColor"/>
</svg>
    </button>
</nav>
<!-- /ko -->

            <!-- main content area - info pages or questions go in here -->
            <main class="mainDisplay" role="main" data-bind="attr: {'aria-labelledby': main_labelledby}">
                <div id="infoDisplay" class="main-content" data-bind="visible: infoPage">
    <div id="frontpage" data-bind="visible: infoPage()=='frontpage', if: infoPage()=='frontpage'">
        <h1 class="break-words" id="infopage-frontpage-header" data-bind="html: exam.settings.name"></h1>
<table class="summary">
    <tbody>
    <!-- ko if: exam.settings.navigateMode != 'diagnostic' -->
    <tr>
        <th data-localise="exam.number of questions"></th>
        <td class="rtd" data-bind="html: numAssessedQuestions"></td>
    </tr>
    <tr data-bind="visible: exam.settings.showTotalMark">
        <th data-localise="exam.marks available"></th>
        <td class="rtd" data-bind="niceNumber: marks"></td>
    </tr>
    <!-- /ko -->
    <tr data-bind="visible: exam.settings.percentPass>0">
        <th data-localise="exam.pass percentage"></th>
        <td class="rtd" data-bind="text: percentPass"></td>
    </tr>
    <tr data-bind="visible: duration()>0">
        <th data-localise="exam.time allowed"></th>
        <td class="rtd"><time data-bind="duration: duration"></time></td>
    </tr>
    <tr data-bind="visible: exam.settings.showStudentName && exam.student_name && !needsStudentName">
        <th data-localise="exam.student name"></th>
        <td><span data-bind="text: exam.student_name"></span><span data-bind="visible: exam.student_id"> (<span data-bind="text: exam.student_id"></span>)</span></td>
    </tr>
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Three examples.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Let&nbsp;<br/>\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}}\\;\\]<br/>Calculate the following $2 \\times 2$ matrices:</p>\n<p>&nbsp;</p>", "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"]}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-5..5 except 0),2),2))", "description": "", "templateType": "anything"}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,1,-1,p1,q])", 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"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "tags": [], "metadata": {"description": "<p>Multiplication of $2 \\times 2$ matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "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    ", "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"]}, "variables": {"ba21": {"name": "ba21", "group": "Ungrouped variables", "definition": "b21*a11+b22*a21", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "ba22": {"name": "ba22", "group": "Ungrouped variables", "definition": "b21*a12+b22*a22", "description": "", "templateType": "anything"}, "cb21": {"name": "cb21", "group": "Ungrouped variables", "definition": "c21*b11+c22*b21", "description": "", "templateType": "anything"}, "b22": {"name": "b22", "group": "Ungrouped variables", "definition": "random(-3..-1)", "description": "", "templateType": "anything"}, "b21": {"name": "b21", "group": "Ungrouped variables", "definition": "random(2,3)", "description": "", "templateType": "anything"}, "cb22": {"name": "cb22", "group": "Ungrouped variables", "definition": "c21*b12+c22*b22", "description": "", "templateType": "anything"}, "ac22": {"name": "ac22", "group": "Ungrouped variables", "definition": "a21*c12+a22*c22", "description": "", "templateType": "anything"}, "ac21": {"name": "ac21", "group": "Ungrouped variables", "definition": "a21*c11+a22*c21", "description": "", "templateType": "anything"}, "ab22": {"name": "ab22", "group": "Ungrouped variables", "definition": "a21*b12+a22*b22", "description": "", "templateType": "anything"}, "ab21": {"name": "ab21", "group": "Ungrouped variables", "definition": "a21*b11+a22*b21", "description": "", "templateType": "anything"}, "b12": {"name": "b12", "group": "Ungrouped variables", "definition": "random(-3..1)", "description": "", "templateType": "anything"}, "b11": {"name": "b11", "group": "Ungrouped variables", "definition": "random(-3,-1,0,3)", "description": "", "templateType": "anything"}, "c12": {"name": "c12", "group": "Ungrouped variables", "definition": "a12+b12", "description": "", "templateType": "anything"}, "c11": {"name": "c11", "group": "Ungrouped variables", "definition": "random(1,0,4)", "description": "", "templateType": "anything"}, "ab12": {"name": "ab12", "group": "Ungrouped variables", "definition": "a11*b12+a12*b22", "description": "", "templateType": "anything"}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(-2,1,2)", "description": "", "templateType": "anything"}, "cb11": {"name": "cb11", "group": "Ungrouped variables", "definition": "c11*b11+c12*b21", "description": "", "templateType": "anything"}, "cb12": {"name": "cb12", "group": "Ungrouped variables", "definition": "c11*b12+c12*b22", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "ab11": {"name": "ab11", "group": "Ungrouped variables", "definition": "a11*b11+a12*b21", "description": "", "templateType": "anything"}, "ba11": {"name": "ba11", "group": "Ungrouped variables", "definition": "b11*a11+b12*a21", "description": "", "templateType": "anything"}, "ba12": {"name": "ba12", "group": "Ungrouped variables", "definition": "b11*a12+b12*a22", "description": "", "templateType": "anything"}, "c22": {"name": "c22", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything"}, "c21": {"name": "c21", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "ac12": {"name": "ac12", "group": "Ungrouped variables", "definition": "a11*c12+a12*c22", "description": "", "templateType": "anything"}, "ac11": {"name": "ac11", "group": "Ungrouped variables", "definition": "a11*c11+a12*c21", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "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"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "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>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "matrix([\n    [ab11,ab12],\n    [ab21,ab22]\n])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "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>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "matrix([\n    [cb11,cb12],\n    [cb21,cb22]\n])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false}], "sortAnswers": false}], "type": "question"}, {"name": "Shaheen's copy of Shaheen's copy of Priyanka's copy of Find determinants and inverses of 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}, {"name": "Priyanka Pandya", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4134/"}], "tags": [], "metadata": {"description": "<p>Find the determinant and inverse of three $2 \\times 2$ invertible matrices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Let</p>\n<p>\\begin{align} \\mathbf{A} &amp;= \\var{a}, &amp; \\mathbf{B} &amp;= \\var{b}, &amp; \\mathbf{C} &amp;= \\var{c} \\end{align}</p>", "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"]}, "variables": {"b22": {"name": "b22", "group": "Ungrouped variables", "definition": "if(tr3*b11=b21*b12,tr3+1,tr3)", "description": "", "templateType": "anything"}, "b21": {"name": "b21", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything"}, "tr2": {"name": "tr2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Unnamed group", "definition": "matrix([ [b11,b12], [b21,b22] ])", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Unnamed group", "definition": "matrix([ [c11,c12], [c21,c22] ])", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "if(tr1*a11=a21*a12,tr1+1,tr1)", "description": "", "templateType": "anything"}, "c21": {"name": "c21", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "tr3": {"name": "tr3", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything"}, "c22": {"name": "c22", "group": "Ungrouped variables", "definition": "if(tr4*c11=c21*c12,tr4+1,tr4)", "description": "", "templateType": "anything"}, "c11": {"name": "c11", "group": "Ungrouped variables", "definition": "random(1,2,4)", "description": "", "templateType": "anything"}, "tr4": {"name": "tr4", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "b11": {"name": "b11", "group": "Ungrouped variables", "definition": "if(a11=tr2,tr2+1,tr2)", "description": "", "templateType": "anything"}, "tr1": {"name": "tr1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Unnamed group", "definition": "matrix([ [a11,a12],[a21,a22] ])", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "b12": {"name": "b12", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}, "c12": {"name": "c12", "group": "Ungrouped variables", "definition": "a12+b12", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything"}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a11", "a12", "a21", "a22", "b11", "b12", "b21", "b22", "c11", "c12", "c21", "c22", "tr1", "tr2", "tr3", "tr4"], "variable_groups": [{"name": "Unnamed group", "variables": ["a", "b", "c"]}], "functions": {"inverse": {"parameters": [["m", "matrix"]], "type": "matrix", "language": "jme", "definition": "matrix([\n    [m[1][1], -m[0][1]],\n    [-m[1][0], m[0][0]]\n])/det(m)"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate the determinants of these matrices.</p>\n<p>$\\mathrm{det}\\left(A\\right) = $&nbsp;[[0]]</p>\n<p>$\\mathrm{det}\\left(B\\right) = $&nbsp;[[1]]</p>\n<p>$\\mathrm{det}\\left(C\\right) = $&nbsp;[[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.5, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(a)", "maxValue": "det(a)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.5, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(b)", "maxValue": "det(b)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.5, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(c)", "maxValue": "det(c)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Find the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "inverse(a)", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\mathbf{B}^{-1} = $&nbsp;[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "inverse(b)", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true}], "sortAnswers": false}], "type": "question"}, {"name": "Shaheen's copy of Shaheen's copy of Andrew's copy of Matrices:  Inverse Matrix Method", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Dunbar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/770/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "tags": [], "metadata": {"description": "<p>Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Write the following equations as a matrix equation <br/>\\[AX=b\\]for a matrix $A$ and column vectors $X$ 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>", "advice": "<h4>a)</h4>\n<p>The equations can be written in the matrix form:</p>\n<p>\\[\\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<h4>b)</h4>\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<h4>c)</h4>\n<p>We have:</p>\n<p>\\[ \\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<h4>d)</h4>\n<p>Note that $AX = b \\Rightarrow X = A^{-1}b$ 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}\\]</p>\n<p>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>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "sc*random(1..9)", "description": "", "templateType": "anything"}, "sc1": {"name": "sc1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "sa1": {"name": "sa1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "sb1": {"name": "sb1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "sb*random(1..9)", "description": "", "templateType": "anything"}, "sb": {"name": "sb", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "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)))))", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "sa*random(1..9)", "description": "", "templateType": "anything"}, "test": {"name": "test", "group": "Ungrouped variables", "definition": "round(a*b1/b)", "description": "", "templateType": "anything"}, "sa": {"name": "sa", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "ab": {"name": "ab", "group": "Ungrouped variables", "definition": "abs(b)", "description": "", "templateType": "anything"}, "inc": {"name": "inc", "group": "Ungrouped variables", "definition": "sa*random(1..9)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "if(test+inc=0,test+inc+1,test+inc)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "sc1*random(1..9)", "description": "", "templateType": "anything"}, "addortakeaway": {"name": "addortakeaway", "group": "Ungrouped variables", "definition": "if(b*b1<0,'add the equation','take away the equation')", "description": "", "templateType": "anything"}, "da": {"name": "da", "group": "Ungrouped variables", "definition": "a*b1-a1*b", "description": "", "templateType": "anything"}, "sc": {"name": "sc", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "addortakeaway", "c", "b", "da", "a1", "sc", "ab", "sb1", "b1", "test", "sb", "sa", "sc1", "sa1", "c1", "inc"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p style=\"text-align: center;\">$A = $&nbsp;[[0]]</p>\n<table style=\"margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr>\n<td rowspan=\"2\">$X = \\;\\;\\Bigg($</td>\n<td>[[1]]</td>\n<td rowspan=\"2\">$\\Bigg)$</td>\n</tr>\n<tr>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: center;\">$b = $&nbsp;[[3]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "matrix([\n    [a,b],\n    [a1,b1]\n])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "y", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "y", "value": ""}]}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "matrix([\n    [c],\n    [c1]\n])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p><span>Find the inverse of $A$. </span></p>\n<ul>\n<li><span>Input all numbers as fractions or integers (not decimals). </span></li>\n<li><span>You do not need to simplify any fractions.</span></li>\n</ul>\n<p>$A^{-1} = $&nbsp;[[1]][[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "matrix([\n    [b1,-b],\n    [-a1,a]\n])", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "1/{da}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate the following:</p>\n<ul>\n<li>Input your answers as integers or fractions (not decimals).</li>\n<li>You do not need to simplify fractions.</li>\n</ul>\n<p>$A^{-1}b = $&nbsp;[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "matrix([{c*b1-c1*b}/{b1*a-a1*b}],[{-c*a1+c1*a}/{b1*a-a1*b}])", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Now solve the equations, inputting your answers as fractions or integers (not decimals).</p>\n<p><br/>$x = \\;\\;$[[0]]</p>\n<p>$y = \\;\\;$[[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{c*b1-c1*b}/{b1*a-a1*b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input as a fraction or an integer, not as a decimal</p>"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.25", "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{-c*a1+c1*a}/{b1*a-a1*b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input as a fraction or an integer, not as a decimal</p>"}, "valuegenerators": []}], "sortAnswers": false}], "type": "question"}, {"name": "Shaheen's copy of Cramers Rule 3x3", "extensions": [], "custom_part_types": [], "resources": ["question-resources/Cramers.PNG"], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}, {"name": "Marisa Llorens", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2155/"}], "tags": [], "metadata": {"description": "<p>Cramers Rule applied to 3 simultaneous equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><img width=\"876\" height=\"362\" src=\"resources/question-resources/Cramers.PNG\"/></p>\n<p>Using Cramer's rule (above), solve these simultaneous equations:</p>\n<p>$\\var{a11}x+\\var{a12}y+\\var{a13}z=\\var{c1}$</p>\n<p>$\\var{a21}x+\\var{a22}y+\\var{a23}z=\\var{c2}$</p>\n<p>$\\var{a31}x+\\var{a32}y+\\var{a33}z=\\var{c3}$</p>\n<p></p>", "advice": "<p>If \\[&nbsp; A=\\left( \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right),\\]</p>\n<p>\\[&nbsp; C=\\left( \\begin{array}{ccc}<br/>c_{1} \\\\&nbsp;c_{2}&nbsp;\\\\c_{3} \\end{array} \\right),\\]</p>\n<p>Cramer's Rule : ${x_1}=\\frac{\\Delta_1}{\\Delta_0}$ ,&nbsp; ${x_2}=\\frac{\\Delta_2}{\\Delta_0}$ , ${x_3}=\\frac{\\Delta_3}{\\Delta_0}$</p>\n<p>Where:\\[ \\Delta_0=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_1=\\left| \\begin{array}{ccc}<br/>c_{1} &amp;&nbsp;a_{12} &amp; a_{13} \\\\c_{2} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ c_{3} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_2=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;c_{1} &amp; a_{13} \\\\a_{21} &amp;&nbsp;c_{2} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;c_{3} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_3=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; c_{1} \\\\a_{21} &amp;&nbsp;a_{22} &amp; c_{2}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; c_{3}\\end{array} \\right|\\]</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p><span class=\"nb\"><span class=\"k\"><span class=\"s\">&nbsp;</span></span></span></p>\n<p></p>", "rulesets": {}, "variables": {"matrixA2": {"name": "matrixA2", "group": "Cramer determinants", "definition": "matrix([a11,c1,a13],[a21,c2,a23],[a31,c3,a33])", "description": "", "templateType": "anything"}, "a33": {"name": "a33", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(0..10 except(a21*a12/a11))", "description": "", "templateType": "anything"}, "matrixA3": {"name": "matrixA3", "group": "Cramer determinants", "definition": "matrix([a11,a12,c1],[a21,a22,c2],[a31,a32,c3])", "description": "", "templateType": "anything"}, "matrixA1": {"name": "matrixA1", "group": "Cramer determinants", "definition": "matrix([c1,a12,a13],[c2,a22,a23],[c3,a32,a33])", "description": "", "templateType": "anything"}, "a31": {"name": "a31", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a32": {"name": "a32", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "a31*x1+a32*x2+a33*x3", "description": "", "templateType": "anything"}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "a21*x1+a22*x2+a23*x3", "description": "", "templateType": "anything"}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a23": {"name": "a23", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "matrixA": {"name": "matrixA", "group": "Ungrouped variables", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "templateType": "anything"}, "a13": {"name": "a13", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "a11*x1+a12*x2+a13*x3", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "x1", "x2", "x3", "c1", "c2", "c3"], "variable_groups": [{"name": "Cramer determinants", "variables": ["matrixA1", "matrixA2", "matrixA3"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is the determinant of $\\var{matrixA}$?</p>\n<p>[[0]]</p>\n<p></p>\n<p>NOTE : IF YOU GET A DETERMINANT OF ZERO FOR THIS ANSWER PLEASE ALERT THE INVIGILATOR.</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA)", "maxValue": "det(matrixA)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate&nbsp;$\\Delta_x=$ [[0]]</p>\n<p>Hence, calculate $x=$&nbsp; [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA1)", "maxValue": "det(matrixA1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{x1}", "maxValue": "{x1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate&nbsp;$\\Delta_y=$[[0]]</p>\n<p>Hence, calculate ${y=}$&nbsp; [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA2)", "maxValue": "det(matrixA2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{x2}", "maxValue": "{x2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p style=\"text-align: left;\">Calculate&nbsp;$\\Delta_z=$[[0]]</p>\n<p>Hence, calculate $z=$&nbsp; [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "det(matrixA3)", "maxValue": "det(matrixA3)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{x3}", "maxValue": "{x3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "type": "question"}]}], "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"action": "warnifunattempted", "message": "<p>You have not answered this question!</p>"}, "preventleave": true, "startpassword": "y1aii"}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "<p>The matrices part of your in-class 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