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Let
\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]
Calculate the determinants of these matrices:
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$det(A) = $ [[0]]
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\n", "gaps": [{"allowFractions": false, "maxValue": "det(C)", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "customName": "", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "showFeedbackIcon": true, "useCustomName": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "det(C)", "marks": "3", "unitTests": [], "variableReplacements": [], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}], "advice": "The determinant of a 3 x 3 matrix
\n\\[A = \\begin{pmatrix} a_{11} \\ a_{12} \\ a_{13} \\\\ a_{21} \\ a_{22} \\ a_{23} \\\\ a_{31} \\ a_{32} \\ a_{33} \\end{pmatrix}\\]
\nis given by
\n\\[det(A) = a_{11}\\left| \\begin{matrix} a_{22} \\ a_{23} \\\\ a_{32} \\ a_{33}\\end{matrix}\\right| - a_{12}\\left| \\begin{matrix} a_{21} \\ a_{23} \\\\ a_{31} \\ a_{33}\\end{matrix}\\right| + a_{13}\\left| \\begin{matrix} a_{21} \\ a_{22} \\\\ a_{31} \\ a_{32}\\end{matrix}\\right| \\]
\n\nThis is one way of finding the determinant of a matrix. We can choose any row or column, provided it corresponds with the sign matrix, to calculate the determinant.
\n\n\\[\\text{Sign matrix} = \\begin{pmatrix}+ \\ - \\ + \\\\ -\\ + \\ - \\\\ + \\ - \\ + \\end{pmatrix} \\]
\n\nFor further information see Section 4 of the Chapter 10 Notes.
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