Mathematics - Matrices Question with Solution | TestHub

$A=\left[\begin{array}{ccc}0& 2q& r\\ p& q& -r\\ p& -q& r\end{array}\right]$ $\begin{array}{rl}\therefore & A\cdot A^T=\left[\begin{array}{ccc}0& 2q& r\\ p& q& -r\\ p& -q& r\end{array}\right]\times \left[\begin{array}{ccc}0& p& p\\ 2q& q& -q\\ r& -r& r\end{array}\right]\\ & =\left[\begin{array}{ccc}4q^2+r^2& 2q^2-r^2& -2q^2+r^2\\ 2q^2-r^2& p^2+q^2+r^2& p^2-q^2-r^2\\ -2q^2+r^2& p^2-q^2-r^2& p^2+q^2+r^2\end{array}\right]\end{array}$ Given, $A{A}^T=I$ $\therefore 4q^2+r^2=p^2+q^2+r^2=1$ $\Rightarrow p^2-3q^2=0$ and $r^2=1-4q^2$ and $2q^2-r^2=0\Rightarrow r^2=2q^2$ $\begin{array}{ll}\therefore & p^2=\frac{1}{2},q^2=\frac{1}{6}\text{ and }{r}^2=\frac{1}{3}\\ \therefore & |p|=\frac{1}{\sqrt{2}}\end{array}$