Question

Calculate the values of the determinants:
$\left|\begin{array}{cccc}0& x& y& z\\ x& 0& z& y\\ y& z& 0& x\\ z& y& x& 0\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{cccc}0& x& y& z\\ x& 0& y& z\\ y& z& 0& x\\ z& y& x& 0\end{array}\right|R_1=R_1+R_2+R_3+R_4\Rightarrow \Delta =\left|\begin{array}{cccc}x+y+z& x+y+z& x+y+z& x+y+z\\ x& 0& y& z\\ y& z& 0& x\\ z& y& x& 0\end{array}\right|\Rightarrow \Delta =(x+y+z)\left|\begin{array}{cccc}1& 1& 1& 1\\ x& 0& y& z\\ y& z& 0& x\\ z& y& x& 0\end{array}\right|R_2=R_2-x{R}_1,R_3=R_3-y{R}_1,R_4=R_4-z{R}_1,\Rightarrow \Delta =(x+y+z)\left|\begin{array}{cccc}1& 1& 1& 1\\ 0& -x& y-x& z-x\\ 0& z-y& -y& x-y\\ 0& y-z& x-z& -z\end{array}\right|$

Expanding along $C_1$

$\Rightarrow \Delta =(x+y+z)\left|\begin{array}{ccc}-x& y-x& z-x\\ z-y& -y& x-y\\ y-z& x-z& -z\end{array}\right|R_2=R_2-R_3\Rightarrow \Delta =(x+y+z)\left|\begin{array}{ccc}-x& y-x& z-x\\ 0& -y+x-z& x-y-z\\ y-z& x-z& -z\end{array}\right|\Rightarrow \Delta =(x+y+z)(x-y-z)\left|\begin{array}{ccc}-x& y-x& z-x\\ 0& 1& 1\\ y-z& x-z& -z\end{array}\right|\Rightarrow \Delta =-(x+y+z)(x-y-z)\left|\begin{array}{ccc}0& 1& 1\\ -x& y-x& z-x\\ y-z& x-z& -z\end{array}\right|\Rightarrow \Delta =(x+y+z)(y+z-x)\left|\begin{array}{ccc}0& 1& 1\\ -x& y-x& z-x\\ y-z& x-z& -z\end{array}\right|R_3=R_3+z{R}_1\Rightarrow \Delta =(x+y+z)(y+z-x)\left|\begin{array}{ccc}0& 1& 1\\ -x& y-x& z-x\\ y-z& x& 0\end{array}\right|R_2=R_2+x{R}_1\Rightarrow \Delta =(x+y+z)(y+z-x)\left|\begin{array}{ccc}0& 1& 1\\ -x& y& z\\ y-z& x& 0\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =(x+y+z)(y+z-x)\{0-1\{0-z(y-z)\}+1(-x^2-y(y-z)\}\Rightarrow \Delta =(x+y+z\left)\right(y+z-x\left)\right(2yz-x^2-y^2-z^2)$