Question

By using properties of determinants Find :
$\left|\begin{array}{ccc}0& a& -b\\ -a& 0& -c\\ b& c& 0\end{array}\right|=0$.

Answer 1

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

L.H.S$\to \left|\begin{array}{ccc}0& a& -b\\ -a& 0& -c\end{array}\right|$

Expanding along $R_1$

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

$=0-a(0+bc)-b(-ac-0)$

$=-abc+abc$

$=0$.

R.H.S.