<s>[INST] For the matrix A as given below, which of them satisfy $A^6=I$? For the matrix A as given below, which of them satisfy $A^6=I$?


*

*$A=\left(\begin{array}{ccc}
\cos\frac{\pi}{4}&\sin\frac{\pi}{4}&0\\
-\sin\frac{\pi}{4}&\cos\frac{\pi}{4}&0\\
0&0&1
\end{array}\right)$

*$A=\left(\begin{array}{ccc}
1&0&0\\
0&cos\frac{\pi}{3}&\sin\frac{\pi}{3}\\
0&-\sin\frac{\pi}{3}&\cos\frac{\pi}{3}
\end{array}\right)$

*$A=\left(\begin{array}{ccc}
\cos\frac{\pi}{6}&0&\sin\frac{\pi}{6}\\
0&1&0\\
-\sin\frac{\pi}{6}&0&\cos\frac{\pi}{6}
\end{array}\right)$

*$A=\left(\begin{array}{ccc}
\cos\frac{\pi}{2}&\sin\frac{\pi}{2}&0\\
-\sin\frac{\pi}{2}&\cos\frac{\pi}{2}&0\\
0&0&1
\end{array}\right)$
I tried and conclude that the option 3 is correct by the technique of solving the equation$A^6=I$ and getting the sixth roots of unity. but i don't know whether it is correct or not. Kindly help me
 [/INST] Let us take the option (3). the matrix is a rotation matrix. The rotation is about the Y-axis. The X and Z axes are rotated by $\frac{\pi}{6}$. So $A^6$ is same as rotating X and Z axes by $\frac{\pi}{6}$ about the Y Axis, sizx times. This is same as rotating X and Z axes by $\pi$ about the Y Axis, the rotation matrix for which has -I.
Can you do the others now?
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