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Question
Evaluate the following:
\[\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
Solution
Let
\[∆ = \begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
\[∆ = \begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
\[ = x^2 y^2 z^2 \begin{vmatrix}0 & x & x \\ y & 0 & y \\ z & z & 0\end{vmatrix} \left[ \text{ Taking }x^2 \text{ common from }C_1 , y^2\text{ common from }C_2\text{ and }z^2\text{ common from }C_3 \right]\]
\[ = x^3 y^3 z^3 \begin{vmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{vmatrix} \left[\text{ Taking x common from }R_1 , y \text{ common from }R_2\text{ and z common from }R_3 \right]\]
\[ = x^3 y^3 z^3 \begin{vmatrix}0 & 0 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & 0\end{vmatrix} \left[\text{ Applying }C_2 \to C_2 - C_3 \right]\]
\[ = x^3 y^3 z^3 \left( 1 + 1 \right) \left[\text{ Expanding along first row }\right]\]
\[ = 2 x^3 y^3 z^3 \]
\[ \therefore ∆ = 2 x^3 y^3 z^3\]
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