Question

If $A=\left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right]$, find x satisfying 0 < x < $\frac{\mathrm{\pi }}{2}$ when A + A^T = I

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right] \\ {A}^T=\left[\begin{array}{cc}\cos x& -\sin x\\ \sin x& \cos x\end{array}\right] \\ \mathrm{Now}, \\ A+A^T=I \\ \Rightarrow \left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right]+\left[\begin{array}{cc}\cos x& -\sin x\\ \sin x& \cos x\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}\cos x+\cos x& \sin x-\sin x\\ -\sin x+\sin x& \cos x+\cos x\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}2\cos x& 0\\ 0& 2\cos x\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow 2\cos x=1 \\ \Rightarrow \cos x=\frac{1}{2} \\ \Rightarrow x=\frac{\mathrm{\pi }}{3} \end{gathered}$$