Question

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

• A. $\begin{pmatrix}{cc}3034 & 3033 \\ -3033 & -3032\end{pmatrix}$
• B. $\begin{pmatrix}3034 & -3033 \\ 3033 & -3032\end{pmatrix}$
• C. $\begin{pmatrix}3033 & 3032 \\ -3032 & -3031\end{pmatrix}$
• D. $\begin{pmatrix}3032 & 3031 \\ -3031 & -3030\end{pmatrix}$

Answer 1

Answer: (A)

$ M =\begin{bmatrix}\frac{5}{2} & \frac{3}{2} \\ \frac{-3}{2} & \frac{-1}{2}\end{bmatrix} $
$ M =\begin{bmatrix}\frac{3}{2}+1 & \frac{3}{2} \\ \frac{-3}{2} & \frac{-3}{2}+1\end{bmatrix} $
$ M = I +\frac{3}{2}\begin{bmatrix}1 & 1 \\ -1 & -1\end{bmatrix} $
Let $ A =\begin{bmatrix}1 & 1 \\ -1 & -1\end{bmatrix}$
$A ^2= \begin{bmatrix}1 & 1 \\ -1 & -1\end{bmatrix}\begin{bmatrix}1 & 1 \\ -1 & -1\end{bmatrix}=\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} $
$ M ^{2022} =\left( I +\frac{3}{2} A \right)^{2022} $
$ = I +3033 A$
$ =\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+3033 \begin{bmatrix}1 & 1 \\ -1 & -1\end{bmatrix} $
$ =\begin{bmatrix}3034 & 3033 \\ -3033 & -3032\end{bmatrix}$