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

We have, $\left| {\begin{array}{cccccccccccccccccccc}0&{x{y^2}}&{x{z^2}}\\{{x^2}y}&0&{y{z^2}}\\{{x^2}z}&{z{y^2}}&0\end{array}} \right| = {x^2}{y^2}{z^2}\left| {\begin{array}{cccccccccccccccccccc}0&x&x\\y&0&y\\z&z&0\end{array}} \right|$

[taking ${x^2},{y^2}$ and ${z^2}$ common from ${C_1},{C_2}$ and ${C_3}$,

respectively]
$= {x^2}{y^2}{z^2}\left| {\begin{array}{cccccccccccccccccccc}0&0&x\\y&{ - y}&y\\z&z&0\end{array}} \right|$

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

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