Question

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. $\left(\begin{array}{cc}cc3034& 3033\\ -3033& -3032\end{array}\right)$
• B. $\left(\begin{array}{cc}3034& -3033\\ 3033& -3032\end{array}\right)$
• C. $\left(\begin{array}{cc}3033& 3032\\ -3032& -3031\end{array}\right)$
• D. $\left(\begin{array}{cc}3032& 3031\\ -3031& -3030\end{array}\right)$

Answer 1

Answer: (A)

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