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

be such that $A +A ^{'}=I$ then the value of $\alpha$ is
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$A=\begin{bmatrix} cos \alpha & sin \alpha \\ -sin \alpha & cos \alpha \end{bmatrix}$

$A^{t}=A^{'}=\begin{bmatrix} cos \alpha & -sin \alpha \\ sin \alpha & cos \alpha \end{bmatrix}$

$A+A^{'}=\begin{bmatrix} cos \alpha & sin \alpha \\ -sin \alpha & cos \alpha \end{bmatrix}+\begin{bmatrix} cos \alpha &- sin \alpha \\ sin \alpha & cos \alpha \end{bmatrix}$

$A+A^{'}=\begin{bmatrix} 2cos \alpha & 0 \\ 0 & 2cos \alpha \end{bmatrix}$

Put $\alpha =\frac{\pi}{3}$

$A+A^{'}=\begin{bmatrix} 2 \times \frac{1}{2} & 0 \\ 0 & 2 \times \frac{1}{2} \end{bmatrix}$

$A+A^{'}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=I$

This is also true for $\frac{5\pi}{3}$ .

The period of the $cos \alpha$ function is $2 \pi$ so values will repeat every $2 \pi$  radians in both directions.

Hence,

$\alpha =\frac{\pi}{3} \pm 2 \pi n, \frac{5\pi }{3} \pm 2 \pi n$
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