Question

If $A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$ and $A^{\prime }+A=I$, then find general solution of $\alpha$

Answer 1

$A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]A^{\prime }=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$

given $A^{\prime }+A=I_2$

$\Rightarrow \left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]+\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}2\cos \alpha & 0\\ 0& 2\cos \alpha \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Comparing the corresponding entries,

we get $2\cos 2\alpha =1$

$\Rightarrow \cos \alpha =1/2$

$\Rightarrow \cos \alpha =\cos \pi /3,\cos 5\pi /3,\cos \frac{7\pi }{3}....$

$\Rightarrow \alpha =\pi /3,\frac{5\pi }{3},\frac{7\pi }{3}....$

$\Rightarrow \alpha =2n\pi \pm \frac{\pi }{3}$