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"customName": "", "prompt": "<p></p>\n<p>Calculate the&nbsp;nine cofactors of $A=\\var{matrixA}$?</p>\n<p>$A _{11}$ cofactor in position 1,1 is [[0]]</p>\n<p>$A_{12}$ cofactor in position 1,2 is [[1]]</p>\n<p>$A_{13}$ cofactor in position 1,3 is [[2]]</p>\n<p>$A_{21}$ cofactor in position 2,1 is [[3]]</p>\n<p>$A_{22}$ cofactor in position 2,2 is [[4]]</p>\n<p>$A_{23}$ cofactor in position 2,3 is [[5]]</p>\n<p>$A_{31}$ cofactor in position 3,1 is [[6]]</p>\n<p>$A_{32}$ cofactor in position 3,2 is [[7]]</p>\n<p>$A_{33}$ cofactor in position 3,3 is [[8]]</p>", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}}, {"useCustomName": false, "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "gaps": [{"unitTests": [], "maxValue": "det(matrixA)", "showFractionHint": true, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "customName": "det a", "allowFractions": false, "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "useCustomName": true, "showCorrectAnswer": true, "marks": "1", "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "mustBeReduced": false, "scripts": {}, "variableReplacements": [], "minValue": "det(matrixA)", "correctAnswerStyle": "plain"}], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "unitTests": [], "customName": "", "prompt": "<p>What is the determinant of $A=\\var{matrixA}$?</p>\n<p>$|A| = $[[0]]</p>", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}}, {"useCustomName": false, "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "gaps": [{"unitTests": [], "scripts": {}, "numColumns": "3", "type": "matrix", "variableReplacementStrategy": "originalfirst", "tolerance": "0.005", "precisionType": "dp", "numRows": "3", "customName": "inv a", "allowFractions": false, "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "extendBaseMarkingAlgorithm": true, "useCustomName": true, "showCorrectAnswer": true, "marks": "9", "markPerCell": true, "precision": "2", "correctAnswer": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/det(matrixA)", "correctAnswerFractions": false, "variableReplacements": [{"variable": "cof11", "must_go_first": true, "part": "p0g0"}, {"variable": "cof12", "must_go_first": true, "part": "p0g1"}, {"variable": "cof13", "must_go_first": true, "part": "p0g2"}, {"variable": "cof21", "must_go_first": true, "part": "p0g3"}, {"variable": "cof22", "must_go_first": true, "part": "p0g4"}, {"variable": "cof23", "must_go_first": true, "part": "p0g5"}, {"variable": "cof31", "must_go_first": true, "part": "p0g6"}, {"variable": "cof32", "must_go_first": true, "part": "p0g7"}, {"variable": "cof33", "must_go_first": true, "part": "p0g8"}, {"variable": "detA", "must_go_first": true, "part": "p1g0"}], "strictPrecision": true, "allowResize": false}], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "unitTests": [], "customName": "", "prompt": "<p>What is the&nbsp;inverse of $A=\\var{matrixA}$? Cofactors will be accepted as fractions or correct to 2 decimal places.</p>\n<p>$A^{-1}=$[[0]]</p>", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "scripts": {}}], "functions": {}, "name": "W1b - Cofactors, Determinant and Inverse of a 3x3 matrix", "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "", "preamble": {"css": "", "js": ""}, "rulesets": {}, "variable_groups": [{"variables": ["cof11", "cof12", "cof13", "cof21", "cof22", "cof23", "cof31", "cof32", "cof33"], "name": "cofactors"}, {"variables": [], "name": "Unnamed group"}], "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= \\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>", "variables": {"inverseA": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/detA", "name": "inverseA"}, "cof22": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a11*a33-a31*a13", "name": "cof22"}, "a12": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(0..10)", "name": "a12"}, "cof11": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a22*a33-a32*a23", "name": "cof11"}, "a32": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(0..10)", "name": "a32"}, "cof23": {"templateType": "anything", "description": "<p>cof23</p>", "group": "cofactors", "definition": "a12*a31-a11*a32", "name": "cof23"}, "a22": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(0..5 except(a21*a12/a11))", "name": "a22"}, "cof32": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a13*a21-a11*a23", "name": "cof32"}, "a21": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(0..10)", "name": "a21"}, "cof13": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a21*a32-a31*a22", "name": "cof13"}, "matrixA": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "name": "matrixA"}, "a31": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(0..10)", "name": "a31"}, "cof21": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a32*a13-a12*a33", "name": "cof21"}, "a13": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-5..10)", "name": "a13"}, "cof12": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a23*a31-a21*a33", "name": "cof12"}, "cof33": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a11*a22-a12*a21", "name": "cof33"}, "cof31": {"templateType": "anything", "description": "", "group": "cofactors", "definition": "a12*a23-a22*a13", "name": "cof31"}, "detA": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "a11*cof11+a12*cof12+a13*cof13", "name": "detA"}, "a23": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-4..4)", "name": "a23"}, "a33": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(0..20)", "name": "a33"}, "a11": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-3..3)", "name": "a11"}}, "metadata": {"description": "<p>Cofactors Determinant and inverse of a 3x3 matrix.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}]}]}], "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}]}