Evaluate $\begin{vmatrix} 0 & xy^2 & xz^2 \\ x^2y & 0 & yz^2 \\ x^2z & zy^2 & 0 \end{vmatrix}$.

$2x^3y^3z^3$

The correct answer is Option (2) → $2x^3y^3z^3$ ##

We have, $\begin{vmatrix} 0 & xy^2 & xz^2 \\ x^2y & 0 & yz^2 \\ x^2z & zy^2 & 0 \end{vmatrix} = x^2 y^2 z^2 \begin{vmatrix} 0 & x & x \\ y & 0 & y \\ z & z & 0 \end{vmatrix}$

[taking $x^2, y^2$ and $z^2$ common from $C_1, C_2$ and $C_3$, respectively]

$= x^2 y^2 z^2 \begin{vmatrix} 0 & 0 & x \\ y & -y & y \\ z & z & 0 \end{vmatrix} \quad [∵C_2 \to C_2 - C_3]$

On expanding along $R_1$, we get

$= x^2 y^2 z^2 [x(yz + yz)]$

$= x^2 y^2 z^2 \cdot 2xyz = 2x^3 y^3 z^3$