1. Tardigrade
  2. Question
  3. Mathematics
  4. Let n ge2 be an integer, A= beginpmatrix cos(2π/ n)& sin (2π / n)&0 - sin(2π / n)& cos(2π / n)&0 0&0&1 endpmatrix and Ι is the identity matrix of order 3. Then

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Q. 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 $Ι$ is the identity matrix of order $3$. Then

WBJEEWBJEE 2014Matrices

Solution:

$A=\begin{bmatrix}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix}$
Now, $A \times A=\begin{bmatrix}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix} \times$
$\begin{bmatrix}\cos \left(\frac{2 \pi}{n}\right) & \sin \left(\frac{2 \pi}{n}\right) & 0 \\ -\sin \left(\frac{2 \pi}{n}\right) & \cos \left(\frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix}$
image
$=\begin{bmatrix}\cos \left(2 \times \frac{2 \pi}{n}\right) & \sin \left(2 \times \frac{2 \pi}{n}\right) & 0 \\ -\sin \left(2 \times \frac{2 \pi}{n}\right) & \cos \left(2 \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix}$
$=\begin{bmatrix}\cos \left(2 \times \frac{2 \pi}{n}\right) & \sin \left(2 \times \frac{2 \pi}{n}\right) & 0 \\ -\sin \left(2 \times \frac{2 \pi}{n}\right) & \cos \left(2 \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix}$
Similarly
$A^{n}=\begin{bmatrix}\cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) & \sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) & 0 \\ -\sin \left(2^{n-1} \times \frac{2 \pi}{n}\right) & \cos \left(2^{n-1} \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix}$
$=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}=I$
and
$A^{n-1}=\begin{bmatrix}\cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) & \sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) & 0 \\ -\sin \left(2^{n-2} \times \frac{2 \pi}{n}\right) & \cos \left(2^{n-2} \times \frac{2 \pi}{n}\right) & 0 \\ 0 & 0 & 1\end{bmatrix}$
$\neq I$
$\therefore A^{m} \neq I$ for any positive integer $m$.