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If \( A=\left[\begin{array}{cc}\cos \alpha+\sin \alpha & \sqrt{2} \sin \alpha \\ -\sqrt{2} \sin \alpha & \cos \alpha-\sin \alpha\end{array}\right] \), prove that
\[
A^{n}=\left[\begin{array}{cc}
\cos n \alpha+\sin n \alpha & \sqrt{2} \sin n \alpha \\
-\sqrt{2} \sin n \alpha & \cos n \alpha-\sin n \alpha
\end{array}\right] \text { for all } n \in N .
\]
\( \mathrm{P} \)
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