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Question
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
Solution
\[ ∆ = \begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
\[ = \frac{xyz}{xyz}\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
\[ = \frac{1}{xyz}\begin{vmatrix}0 & xz & yz \\ - xy & 0 & zy \\ - yx & - zx & 0\end{vmatrix}\]
\[ = \frac{1}{xyz}\begin{vmatrix}- 2xy & 0 & 2yz \\ - xy & 0 & zy \\ - yx & - zx & 0\end{vmatrix} \left[ \text{ Applying } R_1 \to R_1 + R_2 + R_3 \right]\]
\[ = \frac{1}{xyz}\begin{vmatrix}0 & 0 & 0 \\ - xy & 0 & zy \\ - yx & - zx & 0\end{vmatrix} = 0 \left[ \text{ Applying } R_1 \to R_1 - 2 R_2 \right]\]
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