// Numbas version: exam_results_page_options {"name": "Cofactors, Determinant and Inverse of a 3x3 matrix (Adaptive)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "

Cofactors Determinant and inverse of a 3x3 matrix.

", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "statement": "", "variable_groups": [{"variables": ["cof11", "cof12", "cof13", "cof21", "cof22", "cof23", "cof31", "cof32", "cof33"], "name": "cofactors"}, {"variables": [], "name": "Unnamed group"}], "variables": {"detA": {"definition": "a11*cof11+a12*cof12+a13*cof13", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "detA"}, "a31": {"definition": "random(0..10)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a31"}, "cof21": {"definition": "a32*a13-a12*a33", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof21"}, "a21": {"definition": "random(0..10)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a21"}, "a32": {"definition": "random(0..10)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a32"}, "a22": {"definition": "random(0..5 except(a21*a12/a11))", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a22"}, "cof32": {"definition": "a13*a21-a11*a23", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof32"}, "a13": {"definition": "random(-5..10)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a13"}, "a33": {"definition": "random(0..20)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a33"}, "cof33": {"definition": "a11*a22-a12*a21", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof33"}, "cof22": {"definition": "a11*a33-a31*a13", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof22"}, "a11": {"definition": "random(-3..3)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a11"}, "inverseA": {"definition": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/detA", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "inverseA"}, "cof11": {"definition": "a22*a33-a32*a23", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof11"}, "matrixA": {"definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "matrixA"}, "a23": {"definition": "random(-4..4)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a23"}, "cof12": {"definition": "a23*a31-a21*a33", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof12"}, "cof31": {"definition": "a12*a23-a22*a13", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof31"}, "cof23": {"definition": "a12*a31-a11*a32", "group": "cofactors", "description": "

cof23

", "templateType": "anything", "name": "cof23"}, "cof13": {"definition": "a21*a32-a31*a22", "group": "cofactors", "description": "", "templateType": "anything", "name": "cof13"}, "a12": {"definition": "random(0..10)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a12"}}, "rulesets": {}, "functions": {}, "name": "Cofactors, Determinant and Inverse of a 3x3 matrix (Adaptive)", "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "inverseA", "detA"], "tags": [], "parts": [{"customName": "", "prompt": "

\n

Calculate the nine cofactors of A=$\\var{matrixA}$?

\n

$A _{11}$ cofactor in position 1,1[[0]]

\n

$A_{12}$ cofactor in position 1,2[[1]]

\n

$A_{13}$ cofactor in position 1,3[[2]]

\n

$A_{21}$ cofactor in position 2,1[[3]]

\n

$A_{22}$ cofactor in position 2,2[[4]]

\n

$A_{23}$ cofactor in position 2,3[[5]]

\n

$A_{31}$ cofactor in position 3,1[[6]]

\n

$A_{32}$ cofactor in position 3,2[[7]]

\n

$A_{33}$ cofactor in position 3,3[[8]]

What is the determinant of A=$\\var{matrixA}$?

\n

[[0]]

", "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "type": "gapfill", "useCustomName": false, "sortAnswers": false, "gaps": [{"customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "minValue": "det(matrixA)", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "useCustomName": false, "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "showFractionHint": true, "mustBeReduced": false, "maxValue": "det(matrixA)", "customName": "", "variableReplacements": [], "unitTests": [], "allowFractions": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "showCorrectAnswer": true, "marks": "1", "scripts": {}}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "marks": 0, "scripts": {}}, {"customName": "", "prompt": "

\n

What is the inverse of A=$\\var{matrixA}$? Cofactors will be accepted as fractions or correct to 2 decimal places.

\n

[[0]]

", "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "type": "gapfill", "useCustomName": false, "sortAnswers": false, "gaps": [{"precisionMessage": "You have not given your answer to the correct precision.", "numRows": "3", "correctAnswerFractions": false, "precisionPartialCredit": 0, "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "type": "matrix", "numColumns": "3", "showFeedbackIcon": true, "useCustomName": false, "extendBaseMarkingAlgorithm": true, "correctAnswer": "matrix([cof11/detA, cof21/detA, cof31/detA],\n [cof12/detA, cof22/detA, cof32/detA],\n [cof13/detA, cof23/detA, cof33/detA])", "allowResize": false, "customName": "", "markPerCell": false, "variableReplacements": [{"must_go_first": true, "variable": "cof11", "part": "p0g0"}, {"must_go_first": true, "variable": "cof12", "part": "p0g1"}, {"must_go_first": true, "variable": "cof13", "part": "p0g2"}, {"must_go_first": true, "variable": "cof21", "part": "p0g3"}, {"must_go_first": true, "variable": "cof22", "part": "p0g4"}, {"must_go_first": true, "variable": "cof23", "part": "p0g5"}, {"must_go_first": true, "variable": "cof31", "part": "p0g6"}, {"must_go_first": true, "variable": "cof32", "part": "p0g7"}, {"must_go_first": true, "variable": "cof33", "part": "p0g8"}, {"must_go_first": true, "variable": "detA", "part": "p1g0"}], "unitTests": [], "tolerance": "0.005", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "showCorrectAnswer": true, "precision": "2", "marks": 1, "strictPrecision": true, "scripts": {}}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "marks": 0, "scripts": {}}], "advice": "

If \$A=\\left( \\begin{array}{ccc} a & b & c \\\\d & e&f\\\\ g&h&j \\end{array} \\right),\$

\n

Cofactors are given by \$A=\\left( \\begin{array}{ccc} a & b & c \\\\d & e&f\\\\ g&h&j \\end{array} \\right),\$

\n

Cof11 =\$+\\left| \\begin{array}{ccc} e&f\\\\ h&j \\end{array} \\right|,\$

\n

Cof12 =\$-\\left| \\begin{array}{ccc} d & f\\\\ g&j \\end{array} \\right|,\$

\n

Cof13 =\$+\\left| \\begin{array}{ccc} d & e\\ g&h\\end{array} \\right|,\$

\n

Cof21 =\$-\\left| \\begin{array}{ccc} b & c \\\\h&j \\end{array} \\right|,\$

\n

Cof22 =\$+\\left| \\begin{array}{ccc} a & c \\\\ g&j \\end{array} \\right|,\$

\n

Cof23 =\$-\\left| \\begin{array}{ccc} a & b \\\\g&h\\end{array} \\right|,\$

\n

Cof31 =\$+=\\left| \\begin{array}{ccc} b & c \\\\e&f\\end{array} \\right|,\$

\n

Cof32 =\$-\\left| \\begin{array}{ccc} a & c \\\\d & f\\end{array} \\right|,\$

\n

Cof33 =\$+\\left| \\begin{array}{ccc} a & b\\\\d & e \\end{array} \\right|,\$

\n

Then, the determinant of A is given by the sum of the product of any row ( or column) elements by their cofactors

\n

e.g row 1 determinant = a*cof11+b*cof12+c*cof13

\n

and the inverse of A is given by the ratio of the adjoint(A) and the deteminant of A

\n

where adjoint A= \\left( \\begin{array}{ccc}
cof11 & cof21 & cof31 \\\\cof12 & cof22&cof32\\\\ cof13&cof23&cof33 \\end{array} \\right),\\]

\n

inverse of A=\$\\frac{1}{det(A)}*\\left( \\begin{array}{ccc} cof11 & cof21 & cof31 \\\\cof12 & cof22&cof32\\\\ cof13&cof23&cof33 \\end{array} \\right),\$

\n

\n

", "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/"}]}]}], "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/"}]}