JEE Advanced 2022 Paper 2 Q16 Mathematics Matrices & Determinants Matrix Algebra Hard

JEE Advanced 2022 Paper 2 · Q16 · Matrix Algebra

If $M = \begin{pmatrix}\dfrac{5}{2} & \dfrac{3}{2}\\[3pt] -\dfrac{3}{2} & -\dfrac{1}{2}\end{pmatrix}$, then which of the following matrices is equal to $M^{2022}$?

  1. A. $\begin{pmatrix}3034 & 3033\\ -3033 & -3032\end{pmatrix}$
  2. B. $\begin{pmatrix}3034 & -3033\\ 3033 & -3032\end{pmatrix}$
  3. C. $\begin{pmatrix}3033 & 3032\\ -3032 & -3031\end{pmatrix}$
  4. D. $\begin{pmatrix}3032 & 3031\\ -3031 & -3030\end{pmatrix}$
Reveal answer + step-by-step solution

Correct answer:A

Solution

Write $M = I + A$ where $A = M - I = \begin{pmatrix}3/2 & 3/2\\ -3/2 & -3/2\end{pmatrix} = \dfrac{3}{2}\begin{pmatrix}1 & 1\\ -1 & -1\end{pmatrix}$. Note $\begin{pmatrix}1&1\\ -1&-1\end{pmatrix}^2 = \mathbf 0$, so $A^2 = 0$. Hence $(I+A)^n = I + nA$ for all $n\ge 1$. Therefore $M^{2022} = I + 2022 A = I + 2022\cdot\dfrac{3}{2}\begin{pmatrix}1&1\\ -1&-1\end{pmatrix} = I + 3033\begin{pmatrix}1&1\\ -1&-1\end{pmatrix} = \begin{pmatrix}1+3033 & 3033\\ -3033 & 1-3033\end{pmatrix} = \begin{pmatrix}3034 & 3033\\ -3033 & -3032\end{pmatrix}$. Option (A).

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