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Find the determinant of a $3 \\times 3$ matrix.

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Consider the $3 \\times 3$ matrix,

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\\begin{align} \\mathrm{A} &= \\var{a} \\end{align}

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Determinant of a $3 \\times 3$ matrix

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The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&b&c \\\\ d&e&f \\\\ g&h&i \\end{pmatrix}$ can be calculated by using cofactor expansion. Expanding along the first row,

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\\[ \\det\\left(\\mathrm{M}\\right) = a \\cdot \\det \\begin{pmatrix} e&f \\\\ h&i \\end{pmatrix}- b \\cdot \\det \\begin{pmatrix} d&f \\\\ g&i \\end{pmatrix} + c \\cdot \\det \\begin{pmatrix} d&e \\\\ g&h \\end{pmatrix}\\]

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Thus for our example we have:

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\\[\\begin{align} \\det \\begin{pmatrix} e&f \\\\ h&i \\end{pmatrix} &= \\simplify[]{({a22}*{a33})-({a23}*{a32}) = {m1}} \\\\ \\det \\begin{pmatrix} d&f \\\\ g&i \\end{pmatrix} &= \\simplify[]{({a21}*{a33})-({a23}*{a31}) = {m2}} \\\\ \\det \\begin{pmatrix} d&e \\\\ g&h \\end{pmatrix} &=\\simplify[]{ ({a21}*{a32})-({a22}*{a31}) ={m3}}  \\end{align}\\]

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and so

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\\[\\begin{align} \\det\\left(\\mathrm{A}\\right) = (\\simplify[]{{a11}*{m1}})-(\\simplify[]{{a12}*{m2}})+(\\simplify[]{{a13}*{m3}}) = \\simplify[]{{det(a)}}  \\end{align}\\]

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Matrix element

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Matrix element

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Matrix element

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Calculate the determinant of the matrix.

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$\\operatorname{det}\\left( \\mathrm{A}\\right) = $ [[0]]

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