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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:
\[A = \begin{bmatrix}a & h & g \\ h & b & f \\ g & f & c\end{bmatrix}\]
Solution
\[M_{11} = \begin{vmatrix}b & f \\ f & c\end{vmatrix} = bc - f^2 \]
\[ M_{21 =} \begin{vmatrix}h & g \\ f & c\end{vmatrix} = hc - fg\]
\[ M_{31 =} \begin{vmatrix}h & g \\ b & f\end{vmatrix} = hf - gb\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} M_{11} = bc - f^2 \]
\[ C_{21} = \left( - 1 \right)^{2 + 1} M_{11} = - \left( hc - fg \right) = fg - hc\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} M_{11} = hf - gb\]
\[D = a\left( bc - f^2 \right) - h\left( hc - fg \right) + g\left( fh - bg \right)\]
\[ = abc - a f^2 - h^2 c + fgh + fgh - b g^2 \]
\[ = abc + 2hfg - a f^2 - b g^2 - c h^2\]
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