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प्रश्न
Write the minor and cofactor of element of the first column of the following matrix and hence evaluate the determinant:
\[A = \begin{bmatrix}1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2\end{bmatrix}\]
उत्तर
\[M_{11} = \begin{vmatrix}- 1 & 2 \\ 5 & 2\end{vmatrix} = - 2 - 10 = - 12\]
\[ M_{21 =} \begin{vmatrix}- 3 & 2 \\ 5 & 2\end{vmatrix} = - 6 - 10 = - 16\]
\[ M_{31 =} \begin{vmatrix}- 3 & 2 \\ - 1 & 2\end{vmatrix} = - 6 + 2 = - 4\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} M_{11} = - 12\]
\[ C_{21 =} \left( - 1 \right)^{2 + 1} M_{21} = - \left( - 16 \right) = 16\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} M_{31} = - 4\]
\[D = 1\left( - 12 \right) + 3\left( 8 - 6 \right) + 2\left( 20 + 3 \right) = - 12 + 6 + 46 = 40\]
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