$\begin{array}{l} A =\frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right] \\ A =\left[\begin{array}{cc}\cos 60^{\circ} & \sin 60^{\circ} \\ -\sin 60^{\circ} & \cos 60^{\circ}\end{array}\right]\end{array}$

$\text { If } A =\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right] \text { Here } \alpha=\frac{\pi}{3} ~\\ A ^2=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$

$\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right] ~\\ =\left[\begin{array}{cc}\cos 2 \alpha & \sin 2 \alpha \\ -\sin 2 \alpha & \cos 2 \alpha\end{array}\right]$

$A ^{30}=\left[\begin{array}{cc}\cos 30 \alpha & \sin 30 \alpha \\ -\sin 30 \alpha & \cos 30 \alpha\end{array}\right] ~\\ A ^{30}=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]= I$

$A^{25}=\left[\begin{array}{cc}\cos 25 \alpha & \sin 25 \alpha \\ -\sin 25 \alpha & \cos 25 \alpha\end{array}\right]=\left[\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{-\sqrt{3}}{2} & \frac{1}{2}\end{array}\right]$

$A^{25}=A$

$A^{25}-A=0$