JEE Advanced 2022 · Paper 2, Mathematics question 16

Split off the identity.
$$M=I+N,\qquad N=\begin{pmatrix}\frac32&\frac32\\-\frac32&-\frac32\end{pmatrix},\qquad N^2=0.$$

Powers. Since $N^2=0$, $M^n=I+nN$. With $n=2022$, $\frac{3n}2=3033$:
$$M^{2022}=\begin{pmatrix}1+3033&3033\\-3033&1-3033\end{pmatrix}=\begin{pmatrix}3034&3033\\-3033&-3032\end{pmatrix}.$$
(A).