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# Find the value of $x, y, z$ if the matrix. $\begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfy the equation $A^TA =I_3$

$(A)\;x = \pm \frac{1}{\sqrt 2} , y = \pm \frac{1}{\sqrt 6} , z = \pm \frac{1}{\sqrt 3}$
$(B)\;x = \pm \frac{1}{\sqrt 3} , y = \pm \frac{1}{\sqrt 2} , z = \pm \frac{1}{\sqrt 6}$
$(C)\;x = \pm \frac{1}{2} , y = \pm \frac{1}{ 6} , z = \pm \frac{1}{3}$
$(D)\; x = \pm \frac{1}{3} , y = \pm \frac{1}{2} , z = \pm \frac{1}{6}$

$A =\begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}\; \; \; \; \; \; \;$ $\therefore A^T = \begin{bmatrix} 0 & x & x \\ 2y & y & -y \\ z & -z & z \end{bmatrix}$
Given $A^TA = I_3$
$\implies \begin{bmatrix} 0 & x & x \\ 2y & y & -y \\ z & -z & z \end{bmatrix} \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
$\implies \begin{bmatrix} 2x^2 & 0 & 0 \\ 0 & 6y^2 & 0 \\ 0 & 0 & 3z^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
$\implies 2x^2 = 1 \; \; \; or \; \; \; x = \pm \frac{1}{\sqrt 2}$
$6y^2 = 1 \; \; \; \; \; or \; \; \; \; y= \pm \frac{1}{\sqrt 6}$
$3z^2 = 1 \; \; \; \; \; or \; \; \; z = \pm \frac{1}{\sqrt 3}$