JEE Advanced 2022 Paper 2, Mathematics Section 3 Q2: Algebra Of Matrices

Write $M=I+N$, where $N=\left(\begin{array}{c}3/23/2\\ -3/2-3/2\end{array}\right)$. Note that $N$ has rank 1 with $N^2=0$ (since the trace of $N$ is zero and its determinant is zero, so $N$ is nilpotent).

Verifying: $N^2=\left(\begin{array}{c}3/23/2\\ -3/2-3/2\end{array}\right)\left(\begin{array}{c}3/23/2\\ -3/2-3/2\end{array}\right)=\left(\begin{array}{c}00\\ 00\end{array}\right)$.

By the binomial expansion, $M^n=(I+N{)}^n=I+nN$.

For $n=2022$: $M^{2022}=I+2022N=\left(\begin{array}{c}1+30333033\\ -30331-3033\end{array}\right)=\left(\begin{array}{c}30343033\\ -3033-3032\end{array}\right)$.

The answer is option (A).