Find the values of x, y, z if the matrix $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&{2y}&z\\x&y&{ - z}\\x&{ - y}&z\end{array}} \right]$ satisfy the equation $A'A = I.$

.:

Given that, matrix $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&{2y}&z\\x&y&{ - z}\\x&{ - y}&z\end{array}} \right]$ and $A'A = I.$

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

$\Rightarrow$ $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{0 + {x^2} + {x^2}}&{0 + xy - xy}&{0 - zx + zx}\\{0 + xy - xy}&{4{y^2} + {y^2} + {y^2}}&{2yz - yz - yz}\\{0 - zx + xz}&{2yz - zy - zy}&{{z^2} + {z^2} + {z^2}}\end{array}} \right]$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&0\\0&1&0\\0&0&1\end{array}} \right]$

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

Hence, $x = \pm \cfrac{1}{{\sqrt 2 }},y = \pm \cfrac{1}{{\sqrt 6 }},z = \pm \cfrac{1}{{\sqrt 3 }}$