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Question
Evaluate the following determinant:
\[\begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c\end{vmatrix}\]
Solution
\[ ∆ = \begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c\end{vmatrix}\]
\[ = a\begin{vmatrix}b & f \\ f & c\end{vmatrix} - h\begin{vmatrix}h & f \\ g & c\end{vmatrix} + g\begin{vmatrix}h & b \\ g & f\end{vmatrix}\]
\[ = a(bc - f^2 ) - h(hc - fg) + g(hf - gb)\]
\[ = abc - a f^2 - h^2 c + fgh + fgh - g^2 b\]
\[ = abc + 2fgh - a f^2 - c h^2 - b g^2 \]
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