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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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$det (B) =$ [[0]]

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$det(C) =$ [[0]]

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The determinant of a 3 x 3 matrix

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\$A = \\begin{pmatrix} a_{11} \\ a_{12} \\ a_{13} \\\\ a_{21} \\ a_{22} \\ a_{23} \\\\ a_{31} \\ a_{32} \\ a_{33} \\end{pmatrix}\$

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is given by

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\$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| \$

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This 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.

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\$\\text{Sign matrix} = \\begin{pmatrix}+ \\ - \\ + \\\\ -\\ + \\ - \\\\ + \\ - \\ + \\end{pmatrix} \$

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For further information see Section 4 of the Chapter 10 Notes.

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apb

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p2

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q

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apb

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cabc

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Three examples of determinant of 2x2 matrices.

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