The value of
Question:

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

Solution:

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

[Taking $x^{2}, y^{2}$ and $z^{2}$ common from $C_{1}, C_{2}$ and $C_{3}$, respectively]

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

[Applying $C_{2} \rightarrow C_{2}-C_{3}$ ]

$=x^{2} y^{2} z^{2}\left|\begin{array}{ccc}0 & 0 & x \\ y & -y & y \\ z & z & 0\end{array}\right|=x^{2} y^{2} z^{2}(x(y z+y z))$

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

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