Question

Prove that :
$\left|\begin{array}{ccc}0& a& -b\\ -a& 0& -c\\ b& c& 0\end{array}\right|=0$.

Answer 1

LHS $=$$\left|\begin{array}{ccc}0& a& -b\\ -a& 0& -c\\ b& c& 0\end{array}\right|$

Expanding the determinant along $R_1$

$=0\left|\begin{array}{cc}0& c\\ c& 0\end{array}\right|$$-a\left|\begin{array}{cc}0& c\\ c& 0\end{array}\right|$$-b\left|\begin{array}{cc}0& c\\ c& 0\end{array}\right|$

$=0-a\left(bc\right)-b(-ac)=abc-abc$

$=0=$RHS

Hence proved.