Question

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

Answer 1

Given A'A =I.
$\therefore {\left[\begin{array}{ccc}0& 2y& z\\ x& y& -z\\ x& -y& z\end{array}\right]}^{\prime }\left[\begin{array}{ccc}0& 2y& z\\ x& y& -z\\ x& -y& z\end{array}\right]=I\Rightarrow \left[\begin{array}{ccc}0& x& x\\ 2y& y& -y\\ z& -z& z\end{array}\right]\left[\begin{array}{ccc}0& 2y& z\\ x& y& -z\\ x& -y& z\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\Rightarrow \left[\begin{array}{ccc}2x^2& 0& 0\\ 0& 6y^2& 0\\ 0& 0& 3z^2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
On comparing the corresponding elements, we have
$2x^2=1,6y^2=1,3z^2=1\Rightarrow x^2=\frac{1}{2},y^2=\frac{1}{6},z^2=\frac{1}{3}\Rightarrow x=\pm \frac{1}{\sqrt{2}},y=\pm \frac{1}{\sqrt{6}},z=\pm \frac{1}{\sqrt{3}}$