Evaluate the following:

Let $\Delta=\left|\begin{array}{ccc}0 & x y^{2} & x z^{2} \\ x^{2} y & 0 & y z^{2} \\ x^{2} z & z y^{2} & 0\end{array}\right|$

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

$=x^{2} y^{2} z^{2}\left|\begin{array}{ccc}0 & x & x \\ y & 0 & y \\ z & z & 0\end{array}\right| \quad\left[\right.$ Taking $x^{2}$ common from $C_{1}, y^{2}$ common from $C_{2}$ and $z^{2}$ common from $\left.C_{3}\right]$

$=x^{3} y^{3} z^{3}\left|\begin{array}{ccc}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right|$ [Taking $x$ common from $R_{1}, y$ common from $R_{2}$ and $z$ common from $R_{3}$ ]

$=x^{3} y^{3} z^{3}\left|\begin{array}{ccc}0 & 0 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & 0\end{array}\right| \quad$ [Applying $C_{2} \rightarrow C_{2}-C_{3}$ ]

$=x^{3} y^{3} z^{3}(1+1) \quad[$ Expanding along first row $]$

$=2 x^{3} y^{3} z^{3}$

$\therefore \Delta=2 x^{3} y^{3} z^{3}$