QJEEAdv22Ma5860: JEE Advanced 2022 Mathematics - Class 12 JEEAdv22Ma5860 Solution: If , then which of the following matrices is equal to

$A=\left(\begin{array}{cc}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$

$A^{2}=\left(\begin{array}{cc}4 & 3 \\ -3 & -2\end{array}\right)$

$A^{3}=\left(\begin{array}{cc}\frac{11}{2} & \frac{9}{2} \\ -\frac{9}{2} & -\frac{7}{2}\end{array}\right)$

$A^{4}=\left(\begin{array}{cc}7 & 6 \\ -6 & -5\end{array}\right)$

and so on

$A^n=\left(\begin{array}{cc}\frac{3 n}{2}+1 & \frac{3 n}{2} \\ -\frac{3 n}{2} & -\frac{3 n}{2}+1\end{array}\right)$

Now

$\begin{aligned}A^{2022} & =\left(\begin{array}{ll}\frac{3 \times 2022}{2}+1 & \frac{3 \times 2022}{2} \\\frac{-3 \times 2022}{2} & \frac{-3 \times 2022}{2}+1\end{array}\right) \\\\& =\left(\begin{array}{cc}3034 & 3033 \\-3033 & -3032\end{array}\right)\end{aligned}$

$\therefore$ Option $(\mathrm{A})$ is correct