Question

Calculate the values of the determinants:
$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =a\{bc-f^2\}-h\{hc-gf\}+g\{hf-bg\}\Rightarrow \Delta =abc-a{f}^2-c{h}^2+fgh+fgh-b{g}^2\Rightarrow \Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2$