Find the values of x, y, z if the matrix satisfy the equation

It is given that $A=\left[\begin{array}{ccc}0 & 2 y & z \\ x & y & -z \\ x & -y & z\end{array}\right]$

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

Now, $A^{\prime} A=I$

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

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

$\Rightarrow\left[\begin{array}{lll}2 x^{2} & 0 & 0 \\ 0 & 6 y^{2} & 0 \\ 0 & 0 & 3 z^{2}\end{array}\right]=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

On comparing the corresponding elements, we have:

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

$6 y^{2}=1 \Rightarrow y=\pm \frac{1}{\sqrt{6}}$

$3 z^{2}=1 \Rightarrow z=\pm \frac{1}{\sqrt{3}}$