Question

Calculate the values of the determinants:
$\left|\begin{array}{ccc}1& z& -y\\ -z& 1& x\\ y& -x& 1\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{ccc}1& z& -y\\ -z& 1& x\\ y& -x& 1\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =1\{1+x^2\}-z\{-z-xy\}+(-y)\{xz-y\}\Rightarrow \Delta =1+x^2+z^2+xyz-xyz+y^2\Rightarrow \Delta =1+x^2+y^2+z^2$