Question

If $A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$, then the number of values of $\alpha$ $\in (0,\pi )$ satisfying $A+A^T=I$, is
[$I$ is an identity matrix of order $2$ and $P^T$ denotes transpose of matrix $P$.]

Answer 1

Answer: 2

Given, $A+A^T=I$
So, $\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]+\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}2\cos \alpha & 0\\ 0& 2\cos \alpha \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\therefore 2\cos \alpha =1\Rightarrow \cos \alpha =\frac{1}{2}$
$\Rightarrow \alpha =\frac{\pi }{3}\text{ or }\frac{5\pi }{3}$
So, number of values of $\alpha \in (0,\pi )$ are two.