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Let $n \geq 2$ be an integer. $A=\left[\begin{array}{ccc}\cos (2 \pi / n) & \sin (2 \pi / n) & 0 \\ -\sin (2 \pi / n) & \cos (2 \pi / n) & 0 \\ 0 & 0 & 1\end{array}\right]$ and $I$ is the identity matrix of order 3 . Then,
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Verified Answer
The correct answer is:
$A^{n}=I$ and $A^{n-1} \neq I$
Given,
$A=\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
Now, $A \times A=\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right] \times$$\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
$\left[\begin{array}{ccc}\cos ^{2}\left(\frac{2 \pi}{n}\right)-\sin ^{2}\left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) \sin \left(\frac{2 \pi}{n}\right)+\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & 0\\ -\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right)-\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & -\sin ^{2}\left(\frac{2 \pi}{n}\right)+\cos ^{2}\left(\frac{2 \pi}{n}\right) & 0\\ 0 & 0 & 1\end{array}\right]$
$=\left[\begin{array}{cccc}\cos \left(2 \times \frac{2 \pi}{n}\right) & 2 \sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & 0 \\ -2 \sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & \cos \left(2 \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
$=\left[\begin{array}{ccc}
\cos \left(2 \times \frac{2 \pi}{n}\right) & \sin \left(2 \times \frac{2 \pi}{n}\right) & 0 \\
-\sin \left(2 \times \frac{2 \pi}{n}\right) & \cos \left(2 \times \frac{2 \pi}{n}\right) & 0 \\
0 & 0 & 1
\end{array}\right]$
Similarly,
$A^{n}=\left[\begin{array}{cccc}\cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) & \sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) & 0 \\ -\sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) & \cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
$=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=I$
and
$\begin{aligned}
A^{n-1}=&\left[\begin{array}{cccc}
\cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) & \sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) & 0 \\
-\sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) & \cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) & 0 \\
0 & 0 & 1
\end{array}\right] \\
\neq I
\end{aligned}$
$\therefore A^{n-1} \neq I$ for any positive integer $m$
$A=\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
Now, $A \times A=\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right] \times$$\left[\begin{array}{ccc}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
$\left[\begin{array}{ccc}\cos ^{2}\left(\frac{2 \pi}{n}\right)-\sin ^{2}\left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) \sin \left(\frac{2 \pi}{n}\right)+\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & 0\\ -\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right)-\sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & -\sin ^{2}\left(\frac{2 \pi}{n}\right)+\cos ^{2}\left(\frac{2 \pi}{n}\right) & 0\\ 0 & 0 & 1\end{array}\right]$
$=\left[\begin{array}{cccc}\cos \left(2 \times \frac{2 \pi}{n}\right) & 2 \sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & 0 \\ -2 \sin \left(\frac{2 \pi}{n}\right) \cos \left(\frac{2 \pi}{n}\right) & \cos \left(2 \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
$=\left[\begin{array}{ccc}
\cos \left(2 \times \frac{2 \pi}{n}\right) & \sin \left(2 \times \frac{2 \pi}{n}\right) & 0 \\
-\sin \left(2 \times \frac{2 \pi}{n}\right) & \cos \left(2 \times \frac{2 \pi}{n}\right) & 0 \\
0 & 0 & 1
\end{array}\right]$
Similarly,
$A^{n}=\left[\begin{array}{cccc}\cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) & \sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) & 0 \\ -\sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) & \cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$
$=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=I$
and
$\begin{aligned}
A^{n-1}=&\left[\begin{array}{cccc}
\cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) & \sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) & 0 \\
-\sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) & \cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) & 0 \\
0 & 0 & 1
\end{array}\right] \\
\neq I
\end{aligned}$
$\therefore A^{n-1} \neq I$ for any positive integer $m$
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