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{"name": "Ex Sheet 6 Cofactors, Determinant and inverse of a 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "", "extensions": [], "name": "Ex Sheet 6 Cofactors, Determinant and inverse of a 3x3 matrix", "tags": [], "variables": {"matrixA": {"templateType": "anything", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "name": "matrixA", "group": "Ungrouped variables", "description": ""}, "a11": {"templateType": "anything", "definition": "random(-3..3)", "name": "a11", "group": "Ungrouped variables", "description": ""}, "cof12": {"templateType": "anything", "definition": "a23*a31-a21*a33", "name": "cof12", "group": "cofactors", "description": ""}, "cof21": {"templateType": "anything", "definition": "a32*a13-a12*a33", "name": "cof21", "group": "cofactors", "description": ""}, "cof11": {"templateType": "anything", "definition": "a22*a33-a32*a23", "name": "cof11", "group": "cofactors", "description": ""}, "a32": {"templateType": "anything", "definition": "random(0..10)", "name": "a32", "group": "Ungrouped variables", "description": ""}, "cof22": {"templateType": "anything", "definition": "a11*a33-a31*a13", "name": "cof22", "group": "cofactors", "description": ""}, "a31": {"templateType": "anything", "definition": "random(0..10)", "name": "a31", "group": "Ungrouped variables", "description": ""}, "cof23": {"templateType": "anything", "definition": "a12*a31-a11*a32", "name": "cof23", "group": "cofactors", "description": "<p>cof23</p>"}, "a23": {"templateType": "anything", "definition": "random(-4..4)", "name": "a23", "group": "Ungrouped variables", "description": ""}, "a13": {"templateType": "anything", "definition": "random(-5..10)", "name": "a13", "group": "Ungrouped variables", "description": ""}, "a33": {"templateType": "anything", "definition": "random(0..20)", "name": "a33", "group": "Ungrouped variables", "description": ""}, "a21": {"templateType": "anything", "definition": "random(0..10)", "name": "a21", "group": "Ungrouped variables", "description": ""}, "cof33": {"templateType": "anything", "definition": "a11*a22-a12*a21", "name": "cof33", "group": "cofactors", "description": ""}, "inverseA": {"templateType": "anything", "definition": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/det(matrixA)", "name": "inverseA", "group": "Ungrouped variables", "description": ""}, "a12": {"templateType": "anything", "definition": "random(0..10)", "name": "a12", "group": "Ungrouped variables", "description": ""}, "cof31": {"templateType": "anything", "definition": "a12*a23-a22*a13", "name": "cof31", "group": "cofactors", "description": ""}, "a22": {"templateType": "anything", "definition": "random(0..5 except(a21*a12/a11))", "name": "a22", "group": "Ungrouped variables", "description": ""}, "cof32": {"templateType": "anything", "definition": "a13*a21-a11*a23", "name": "cof32", "group": "cofactors", "description": ""}, "cof13": {"templateType": "anything", "definition": "a21*a32-a31*a22", "name": "cof13", "group": "cofactors", "description": ""}}, "rulesets": {}, "parts": [{"type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof11}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": true, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof11}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof12}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof12}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof13}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof13}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof21}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof21}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof22}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof22}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof23}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof23}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof31}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof31}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof32}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof32}"}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "{cof33}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "{cof33}"}], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "<p></p>\n<p>Calculate the&nbsp;nine cofactors of&nbsp;A=$\\var{matrixA}$?</p>\n<p>$A _{11}$ cofactor in position 1,1[[0]]</p>\n<p>$A_{12}$ cofactor in position 1,2[[1]]</p>\n<p>$A_{13}$ cofactor in position 1,3[[2]]</p>\n<p>$A_{21}$ cofactor in position 2,1[[3]]</p>\n<p>$A_{22}$ cofactor in position 2,2[[4]]</p>\n<p>$A_{23}$ cofactor in position 2,3[[5]]</p>\n<p>$A_{31}$ cofactor in position 3,1[[6]]</p>\n<p>$A_{32}$ cofactor in position 3,2[[7]]</p>\n<p>$A_{33}$ cofactor in position 3,3[[8]]</p>", "marks": 0}, {"type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "maxValue": "det(matrixA)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "det(matrixA)"}], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "<p>What is the determinant of A=$\\var{matrixA}$?</p>\n<p>[[0]]</p>", "marks": 0}, {"type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "scripts": {}, "allowFractions": true, "numColumns": "3", "variableReplacementStrategy": "originalfirst", "markPerCell": false, "variableReplacements": [], "allowResize": true, "type": "matrix", "numRows": "3", "tolerance": "0.005", "showFeedbackIcon": true, "correctAnswer": "inverseA", "marks": 1, "correctAnswerFractions": false}], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "<p></p>\n<p>What is the&nbsp;inverse of A=$\\var{matrixA}$?</p>\n<p>[[0]]</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Cofactors Determinant and inverse of a 3x3 matrix.</p>"}, "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "inverseA"], "variable_groups": [{"variables": ["cof11", "cof12", "cof13", "cof21", "cof22", "cof23", "cof31", "cof32", "cof33"], "name": "cofactors"}, {"variables": [], "name": "Unnamed group"}], "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "<p>If \\[&nbsp; A=\\left( \\begin{array}{ccc}<br/>a &amp; b &amp; c \\\\d &amp; e&amp;f\\\\ g&amp;h&amp;j \\end{array} \\right),\\]</p>\n<p>Cofactors are given by \\[&nbsp; A=\\left( \\begin{array}{ccc}<br/>a &amp; b &amp; c \\\\d &amp; e&amp;f\\\\ g&amp;h&amp;j \\end{array} \\right),\\]</p>\n<p>Cof11 =\\[&nbsp; +\\left| \\begin{array}{ccc}<br/>e&amp;f\\\\ h&amp;j \\end{array} \\right|,\\]</p>\n<p>Cof12 =\\[&nbsp; -\\left| \\begin{array}{ccc}<br/>d &amp; f\\\\ g&amp;j \\end{array} \\right|,\\]</p>\n<p>Cof13 =\\[&nbsp; +\\left| \\begin{array}{ccc}<br/>d &amp; e\\ g&amp;h\\end{array} \\right|,\\]</p>\n<p>Cof21 =\\[ -\\left| \\begin{array}{ccc}<br/>b &amp; c \\\\h&amp;j \\end{array} \\right|,\\]</p>\n<p>Cof22 =\\[&nbsp; +\\left| \\begin{array}{ccc}<br/>a&nbsp; &amp; c \\\\ g&amp;j \\end{array} \\right|,\\]</p>\n<p>Cof23 =\\[&nbsp; -\\left| \\begin{array}{ccc}<br/>a &amp; b \\\\g&amp;h\\end{array} \\right|,\\]</p>\n<p>Cof31 =\\[&nbsp; +=\\left| \\begin{array}{ccc}<br/>b &amp; c \\\\e&amp;f\\end{array} \\right|,\\]</p>\n<p>Cof32 =\\[ -\\left| \\begin{array}{ccc}<br/>a&nbsp; &amp; c \\\\d &amp; f\\end{array} \\right|,\\]</p>\n<p>Cof33 =\\[&nbsp; +\\left| \\begin{array}{ccc}<br/>a &amp; b\\\\d &amp; e \\end{array} \\right|,\\]</p>\n<p>Then, the determinant of A is given by the sum of the product of any row ( or column) elements by their cofactors </p>\n<p>e.g row 1 determinant = a*cof11+b*cof12+c*cof13</p>\n<p>and the inverse of A is given by the ratio of the adjoint(A) and the deteminant of A</p>\n<p>where adjoint A= \\left( \\begin{array}{ccc}<br/>cof11 &amp; cof21 &amp; cof31 \\\\cof12 &amp; cof22&amp;cof32\\\\ cof13&amp;cof23&amp;cof33 \\end{array} \\right),\\]</p>\n<p><span class=\"nb\">&nbsp; inverse of A=\\[&nbsp; \\frac{1}{det(A)}*\\left( \\begin{array}{ccc}<br/>cof11 &amp; cof21 &amp; cof31 \\\\cof12 &amp; cof22&amp;cof32\\\\ cof13&amp;cof23&amp;cof33 \\end{array} \\right),\\]</span></p>\n<p><span class=\"nb\"><span class=\"k\"><span class=\"s\">&nbsp;</span></span></span></p>\n<p></p>", "type": "question", "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}]}]}], "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}]}