Question

The value of the determinant $∆=\left|\begin{array}{ccc}0& x-y& y-z\\ y-x& 0& z-x\\ z-y& x-z& 0\end{array}\right|$ is _____________.

Answer 1

Given: $∆=\left|\begin{array}{ccc}0& x-y& y-z\\ y-x& 0& z-x\\ z-y& x-z& 0\end{array}\right|$

$$\begin{gathered} \mathrm{Let}A=\left[\begin{array}{ccc}0& x-y& y-z\\ y-x& 0& z-x\\ z-y& x-z& 0\end{array}\right] \\ {A}^T={\left[\begin{array}{ccc}0& x-y& y-z\\ y-x& 0& z-x\\ z-y& x-z& 0\end{array}\right]}^T \\ =\left[\begin{array}{ccc}0& y-x& z-y\\ x-y& 0& x-z\\ y-z& z-x& 0\end{array}\right] \\ =\left(-1\right)\left[\begin{array}{ccc}0& x-y& y-z\\ y-x& 0& z-x\\ z-y& x-z& 0\end{array}\right] \\ =-A \\ \mathrm{Since},A^T=-A \\ \mathrm{Therefore},A\mathrm{is}\mathrm{a}\mathrm{skew}\mathrm{symmetric}\mathrm{matrix}. \\ \mathrm{Thus},\left|A\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}0& x-y& y-z\\ y-x& 0& z-x\\ z-y& x-z& 0\end{array}\right|=0 \\ \Rightarrow ∆=0 \end{gathered}$$

Hence, the value of the determinant $∆=\left|\begin{array}{ccc}0& x-y& y-z\\ y-x& 0& z-x\\ z-y& x-z& 0\end{array}\right|$ is 0.