Question

$A=\begin{bmatrix}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} & 0 \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1\end{bmatrix}$
$B=\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \frac{\pi}{3} & \sin \frac{\pi}{3} \\ 0 & -\sin \frac{\pi}{3} & \cos \frac{\pi}{3}\end{bmatrix}$
$C=\begin{bmatrix}\cos \frac{\pi}{6} & 0 & \sin \frac{\pi}{6} \\ 0 & 1 & 0 \\ -\sin \frac{\pi}{6} & \cos \frac{\pi}{6} & 0\end{bmatrix}$
$D=\begin{bmatrix}\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\ -\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\ 0 & 0 & 1\end{bmatrix}$

• A. $A^{2020}=I$
• B. $B^{2020}=I$
• C. $A D^{2019}=I$
• D. $B^{2022}=I$

Answer 1

Answer: (D)

Given matrix $A=\begin{bmatrix} \cos \frac{\pi}{4} & \sin \frac{\pi}{4} & 0 \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1 \end{bmatrix}$
$\Rightarrow A^{2}=\begin{bmatrix}\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\ -\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\ 0 & 0 & 1\end{bmatrix}$
$\therefore A^{2}=D=\begin{bmatrix}\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\ -\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\ 0 & 0 & 1\end{bmatrix}=\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$
$\therefore A^{4}=\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}=D^{2}$
$\Rightarrow A^{8}=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$
$\therefore A^{2020}=\left(A^{505}\right)^{4}=\left(\left(A^{8}\right)^{63} \cdot A\right)^{4}=\left(I^{63} \cdot A\right)^{4}=A^{4} \neq I$
Similarly, $D^{4}=I$, SO $D^{2019} \neq I$
Now, $B=\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \frac{\pi}{3} & \sin \frac{\pi}{3} \\ 0 & -\sin \frac{\pi}{3} & \cos \frac{\pi}{3}\end{bmatrix}$
$\Rightarrow B^{2}=\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \frac{2 \pi}{3} & \sin \frac{2 \pi}{3} \\ 0 & -\sin \frac{2 \pi}{3} & \cos \frac{2 \pi}{3}\end{bmatrix}$
$\Rightarrow B^{4}=\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \frac{4 \pi}{3} & \sin \frac{4 \pi}{3} \\ 0 & -\sin \frac{4 \pi}{3} & \cos \frac{4 \pi}{3}\end{bmatrix}$
$\Rightarrow B^{6}=\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \frac{6 \pi}{3} & \sin \frac{6 \pi}{3} \\ 0 & -\sin \frac{6 \pi}{3} & \cos \frac{6 \pi}{3}\end{bmatrix}=I$
$\therefore B^{2020} \neq I$ but $B^{2022}=\left(B^{6}\right)^{337}=I^{337}=I$