// Numbas version: finer_feedback_settings {"name": "Matt's copy of Ex 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": [{"variables": {"cof22": {"name": "cof22", "definition": "a11*a33-a31*a13", "group": "cofactors", "description": "", "templateType": "anything"}, "cof23": {"name": "cof23", "definition": "a12*a31-a11*a32", "group": "cofactors", "description": "

cof23

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

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Cofactors Determinant and inverse of a 3x3 matrix.

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

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

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

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

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

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

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

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

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

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

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

\n

[[0]]

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

\n

[[0]]

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