// Numbas version: exam_results_page_options {"name": "Determinant of a 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Determinant of a 3x3 matrix", "advice": "

Given the matrix:

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\$$\\mathbf{A}=\\begin{pmatrix} {a11}&{a12}&{a13}\\\\ {a21}&{a22}&{a23}\\\\{a31}&{a32}&{a33} \\end{pmatrix}\$$

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The determinant of a 3x3 matrix is determined by the formula:

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\$$|\\mathbf{A}|={a11}*\\begin{vmatrix}{a22}&{a23}\\\\{a32}&{a33}\\end{vmatrix}-{a12}*\\begin{vmatrix}{a21}&{a23}\\\\{a31}&{a33}\\end{vmatrix}+{a13}*\\begin{vmatrix}{a21}&{a22}\\\\{a31}&{a32}\\end{vmatrix}\$$

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So in this example:

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\$$\\mathbf{A}=\\begin{pmatrix}x&\\var{a}&\\var{b}\\\\ \\var{c}&x&\\var{d}\\\\\\var{e1}&\\var{f}&\\var{g} \\end{pmatrix}\$$

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\$$|\\mathbf{A}|=x*\\begin{vmatrix}x&\\var{d}\\\\\\var{f}&\\var{g}\\end{vmatrix}-\\var{a}*\\begin{vmatrix}\\var{c}&\\var{d}\\\\\\var{e1}&\\var{g}\\end{vmatrix}+\\var{b}*\\begin{vmatrix}\\var{c}&x\\\\\\var{e1}&\\var{f}\\end{vmatrix}\$$

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\$$|\\mathbf{A}|=x*(x*\\var{g}-\\var{d}*\\var{f})-\\var{a}*(\\var{c}*\\var{g}-\\var{e1}*\\var{d})+\\var{b}*(\\var{c}*\\var{f}-\\var{e1}x)\$$

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\$$|\\mathbf{A}|=x(x-\\simplify{{d}{f}})-\\var{a}(\\simplify{{c}*{g}}-\\simplify{{e1}*{d}})+\\var{b}(\\simplify{{c}*{f}}-\\var{e1}x)\$$

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\$$|\\mathbf{A}|=x^2-\\simplify{{d}*{f}}x-\\simplify{{a}*{c}*{g}}+\\simplify{{a}*{e1}*{d}}+\\simplify{{b}*{c}*{f}}-\\simplify{{b}*{e1}}x\$$

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\$$|\\mathbf{A}|=x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}}\$$

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If  \$$|\\mathbf{A}|=\\var{k}\$$

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\$$x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}}=\\var{k}\$$

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\$$x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{b}*{e1}*{f}*{d}}=0\$$

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This can be solved by formula or by finding factors

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\$$(x-\\simplify{{f}*{d}})(x-\\simplify{{b}*{e1}})=0\$$

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\$$x-\\simplify{{f}*{d}}=0\\,\\,\\,\\,\\,or\\,\\,\\,\\,\\,x-\\simplify{{b}*{e1}}=0\$$

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\$$x=\\simplify{{f}*{d}}\\,\\,\\,\\,\\,or\\,\\,\\,\\,\\,x=\\simplify{{b}*{e1}}\$$

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Calculate the two values for \$$x\$$ that satisfy the equation \$$|A|=\\var{k}\$$

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where the matrix \$$A\$$ is given by:

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\$$\\mathbf{A}=\\begin{pmatrix} x&\\var{a}&\\var{b}\\\\ \\var{c}&x&\\var{d}\\\\\\var{e1}&\\var{f}&\\var{g} \\end{pmatrix}\$$

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Enter the smaller of the two values

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\$$x=\$$ [[0]]

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Enter the larger of the two values

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\$$x=\$$ [[1]]

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This question tests learner's knowledge of the inverse matrix method for a 3x3 matrix.

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