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Question
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1\end{bmatrix}\]
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Solution
\[M_{11} = \begin{vmatrix}5 & 0 \\ 7 & 1\end{vmatrix} = 5 - 0 = 5\]
\[ M_{21} = \begin{vmatrix}2 & 6 \\ 7 & 1\end{vmatrix} = 2 - 42 = - 40\]
\[ M_{31} = \begin{vmatrix}2 & 6 \\ 5 & 0\end{vmatrix} = 0 - 30 = - 30\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} M_{11} = 5\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} M_{21} = - \left( - 40 \right)\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} M_{31} = - 30\]
\[D = 0\left( 5 - 0 \right) - 2\left( 1 - 0 \right) + 6\left( 7 - 15 \right) = - 2 - 48 = - 50\]
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