Question

Evaluate:
$\left|\begin{array}{ccc}0& x{y}^2& x{z}^2\\ x^{2}y& 0& y{z}^2\\ x^{2}z& z{y}^2& 0\end{array}\right|$

Answer 1

We have, $\left|\begin{array}{ccc}0& x{y}^2& x{z}^2\\ x^{2}y& 0& y{z}^2\\ x^{2}z& z{y}^2& 0\end{array}\right|=x^2{y}^2{z}^2\left|\begin{array}{ccc}0& x& x\\ y& 0& y\\ z& z& 0\end{array}\right|$
[taking $x^2,y^2$ and $z^2$ common from $C_1$ , $C_2$ and $C_3$ respectively]
$=x^2{y}^2{z}^2\left|\begin{array}{ccc}0& 0& x\\ y& -y& y\\ z& z& 0\end{array}\right|$ $[\because C_2\to C_2-C_3]$
$=x^2{y}^2{z}^2\left[x\right(yz+yz\left)\right]=x^2{y}^2{z}^2.2xyz=2x^3{y}^3{z}^3$