If $M = \left(\begin{array}{cc}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$?

(A) (A) $\left(\begin{array}{cc}3034 & 3033 \\ -3033 & -3032\end{array}\right)$

(B) (B) $\left(\begin{array}{ll}3034 & -3033 \\ 3033 & -3032\end{array}\right)$

(C) (C) $\left(\begin{array}{cc}3033 & 3032 \\ -3032 & -3031\end{array}\right)$

(D) (D) $\left(\begin{array}{cc}3032 & 3031 \\ -3031 & -3030\end{array}\right)$

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Matrices

M = (\frac{5}{2} - \frac{3}{2} \frac{3}{2} - \frac{1}{2})

, then which of the following matrices is equal to

M^{2022}

Answer: (A)

Rewrite the matrix:

M = [\frac{3}{2} + 1 - \frac{3}{2} \frac{3}{2} - \frac{3}{2} + 1] = I + \frac{3}{2} [1 - 1 1 - 1]

Let

A = [1 - 1 1 - 1]

.

Compute

A^{2}

A^{2} = [1 - 1 1 - 1] [1 - 1 1 - 1] = [0000]

Since

A^{2} = 0

, using the binomial theorem:

M^{2022} = (I + \frac{3}{2} A)^{2022} = I + 2022 \cdot \frac{3}{2} A = I + 3033 A

= [1001] + 3033 [1 - 1 1 - 1] = [3034 - 3033 3033 - 3032]

Therefore, the answer is (A).

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Single correct option