Question

Let $n\ge2$ be an integer, $A=\begin{pmatrix}\cos\left(2\pi/ n\right)&\sin \left(2\pi / n\right)&0\\\ -\sin\left(2\pi / n\right)&\cos\left(2\pi / n\right)&0\\\ 0&0&1\end{pmatrix}$ and $?$ is the identity matrix of order $3$. Then

• A. $A^{n}=I$ and $A^{n-1} \ne I$
• B. $A^{m} \ne I$ for any positive integer $m$
• C. $A$ is not invertible
• D. $A^m = 0$ for a positive integer $m$

Answer 1

Answer: (A)

$A^{n}=I$ and $A^{n-1} \ne I$

Answer 2

$A=\begin{bmatrix}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\\ 0 & 0 & 1\end{bmatrix}$