Find x, y, z if A = satisfies A¢ = A-1.

Matrix A is such that A¢ = A-1

AA¢ = I

$\left[\begin{array}{ccc}0 & 2 y & z \\ x & y & -z \\ x & -y & z\end{array}\right]\left[\begin{array}{ccc}0 & x & x \\ 2 y & y & -y \\ z & -z & z\end{array}\right]=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$\left[\begin{array}{ccc}4 y^{2}+z^{2} & 2 y^{2}-z^{2} & -2 y^{2}+z^{2} \\ 2 y^{2}-z^{2} & x^{2}+y^{2}+z^{2} & x^{2}-y^{2}-z^{2} \\ -2 y^{2}+z^{2} & x^{2}-y^{2}+z^{2} & x^{2}+y^{2}+z^{2}\end{array}\right]=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$4 y^{2}+z^{2}=1$

$2 y^{2}-z^{2}=0$

$x^{2}+y^{2}+z^{2}=1$

$x^{2}-y^{2}-z^{2}=0$

$y^{2}=1 / 6, z^{2}=1 / 3, x^{2}=1 / 2$

So, the roots are:

$x=\pm \frac{1}{\sqrt{2}}$

$y=\pm \frac{1}{\sqrt{6}}$

And,

$z=\pm \frac{1}{\sqrt{3}}$