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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:
\[A = \begin{bmatrix}2 & - 1 & 0 & 1 \\ - 3 & 0 & 1 & - 2 \\ 1 & 1 & - 1 & 1 \\ 2 & - 1 & 5 & 0\end{bmatrix}\]
Solution
\[M_{11} = 0\left( 0 - 5 \right) - 1\left( 0 + 1 \right) - 2\left( 5 - 1 \right) = - 1 - 8 = - 9\]
\[ M_{21} = - 1\left( 0 - 5 \right) + 1(5 - 1) = 5 + 4 = 9\]
\[ M_{31} = - 1\left( 0 + 10 \right) + 1(0 + 1) = - 10 + 1 = - 9\]
\[ M_{41} = - 1(1 - 2) + 1\left( 0 - 1 \right) = 1 - 1 = 0\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} M_{11} = - 9\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} M_{21} = \left( - 1 \right) \times 9\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} M_{31} = - 9\]
\[ C_{41} = \left( - 1 \right)^{4 + 1} M_{41} = 0\]
\[D = 2\begin{vmatrix}0 & 1 & - 2 \\ 1 & - 1 & 1 \\ - 1 & 5 & 0\end{vmatrix} + 1\begin{vmatrix}- 3 & 1 & - 2 \\ 1 & - 1 & 1 \\ 2 & 5 & 0\end{vmatrix} - 1\begin{vmatrix}- 3 & 0 & 1 \\ 1 & 1 & - 1 \\ 2 & - 1 & 5\end{vmatrix}\]
\[ = - 18 - 27 + 15 = 30\]
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