Question

If $A=\begin{bmatrix}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}$, 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

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