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{"name": "W1b - Find determinants and inverses of 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"useCustomName": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customName": "", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "gaps": [{"showFeedbackIcon": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "scripts": {}, "marks": "1", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customName": "det a", "customMarkingAlgorithm": "", "maxValue": "det(a)", "unitTests": [], "mustBeReduced": false, "useCustomName": true, "variableReplacements": [], "allowFractions": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "det(a)", "adaptiveMarkingPenalty": 0}, {"showFeedbackIcon": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "scripts": {}, "marks": "1", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customName": "det b", "customMarkingAlgorithm": "", "maxValue": "det(b)", "unitTests": [], "mustBeReduced": false, "useCustomName": true, "variableReplacements": [], "allowFractions": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "det(b)", "adaptiveMarkingPenalty": 0}, {"showFeedbackIcon": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "scripts": {}, "marks": "1", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customName": "det c", "customMarkingAlgorithm": "", "maxValue": "det(c)", "unitTests": [], "mustBeReduced": false, "useCustomName": true, "variableReplacements": [], "allowFractions": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "det(c)", "adaptiveMarkingPenalty": 0}, {"showFeedbackIcon": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "scripts": {}, "marks": "1", "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "customName": "det abc", "customMarkingAlgorithm": "", "maxValue": "det(a*b*c)", "unitTests": [], "mustBeReduced": false, "useCustomName": true, "variableReplacements": [], "allowFractions": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "det(a*b*c)", "adaptiveMarkingPenalty": 0}], "sortAnswers": false, "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) = $&nbsp;[[2]]</p>\n<p>$\\mathrm{det}\\left(ABC\\right) = $&nbsp;[[3]]</p>", "variableReplacements": []}, {"useCustomName": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customName": "", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "gaps": [{"useCustomName": true, "showFeedbackIcon": true, "tolerance": 0, "numRows": "2", "showCorrectAnswer": true, "scripts": {}, "marks": "4", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "matrix", "allowFractions": true, "customName": "inv a", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "correctAnswer": "inverse(a)", "markPerCell": true, "unitTests": [], "correctAnswerFractions": true, "allowResize": false, "numColumns": "2", "variableReplacements": []}], "sortAnswers": false, "unitTests": [], "prompt": "<p>Find the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals ($1/2$ rather than $0.5$).</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "variableReplacements": []}, {"useCustomName": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customName": "", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "gaps": [{"useCustomName": true, "showFeedbackIcon": true, "tolerance": 0, "numRows": "2", "showCorrectAnswer": true, "scripts": {}, "marks": "4", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "matrix", "allowFractions": true, "customName": "inv b", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "correctAnswer": "inverse(b)", "markPerCell": true, "unitTests": [], "correctAnswerFractions": true, "allowResize": false, "numColumns": "2", "variableReplacements": []}], "sortAnswers": false, "unitTests": [], "prompt": "<p>$\\mathbf{B}^{-1} = $&nbsp;[[0]]</p>", "variableReplacements": []}, {"useCustomName": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customName": "", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "gaps": [{"useCustomName": true, "showFeedbackIcon": true, "tolerance": 0, "numRows": "2", "showCorrectAnswer": true, "scripts": {}, "marks": "4", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "matrix", "allowFractions": true, "customName": "inv c", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "correctAnswer": "inverse(c)", "markPerCell": true, "unitTests": [], "correctAnswerFractions": true, "allowResize": false, "numColumns": "2", "variableReplacements": []}], "sortAnswers": false, "unitTests": [], "prompt": "<p>$\\mathbf{C}^{-1} = $&nbsp;[[0]]</p>", "variableReplacements": []}], "variable_groups": [{"variables": ["a", "b", "c"], "name": "Unnamed group"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the determinant and inverse of three $2 \\times 2$ invertible matrices.</p>"}, "functions": {"inverse": {"definition": "matrix([\n    [m[1][1], -m[0][1]],\n    [-m[1][0], m[0][0]]\n])/det(m)", "parameters": [["m", "matrix"]], "language": "jme", "type": "matrix"}}, "ungrouped_variables": ["a11", "a12", "a21", "a22", "b11", "b12", "b21", "b22", "c11", "c12", "c21", "c22", "tr1", "tr2", "tr3", "tr4"], "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>", "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "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>", "name": "W1b - Find determinants and inverses of 2x2 matrices", "variables": {"c": {"group": "Unnamed group", "definition": "matrix([ [c11,c12], [c21,c22] ])", "description": "", "templateType": "anything", "name": "c"}, "c21": {"group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "name": "c21"}, "tr2": {"group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "name": "tr2"}, "c11": {"group": "Ungrouped variables", "definition": "random(1,2,4)", "description": "", "templateType": "anything", "name": "c11"}, "a": {"group": "Unnamed group", "definition": "matrix([ [a11,a12],[a21,a22] ])", "description": "", "templateType": "anything", "name": "a"}, "b12": {"group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "name": "b12"}, "b21": {"group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "name": "b21"}, "tr3": {"group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "name": "tr3"}, "a11": {"group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "name": "a11"}, "b11": {"group": "Ungrouped variables", "definition": "if(a11=tr2,tr2+1,tr2)", "description": "", "templateType": "anything", "name": "b11"}, "tr4": {"group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "name": "tr4"}, "b22": {"group": "Ungrouped variables", "definition": "if(tr3*b11=b21*b12,tr3+1,tr3)", "description": "", "templateType": "anything", "name": "b22"}, "b": {"group": "Unnamed group", "definition": "matrix([ [b11,b12], [b21,b22] ])", "description": "", "templateType": "anything", "name": "b"}, "a22": {"group": "Ungrouped variables", "definition": "if(tr1*a11=a21*a12,tr1+1,tr1)", "description": "", "templateType": "anything", "name": "a22"}, "c22": {"group": "Ungrouped variables", "definition": "if(tr4*c11=c21*c12,tr4+1,tr4)", "description": "", "templateType": "anything", "name": "c22"}, "a12": {"group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "name": "a12"}, "a21": {"group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "name": "a21"}, "c12": {"group": "Ungrouped variables", "definition": "a12+b12", "description": "", "templateType": "anything", "name": "c12"}, "tr1": {"group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "name": "tr1"}}, "preamble": {"css": "", "js": ""}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "extensions": [], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}]}