If \(A=\left[\begin{array}{ll}3&-4\\ 1&-1\end{array}\right]\) then which of the following is true

\(A^{2022}=\left[\begin{array}{ll}4045 &-8088 \\ 2022 & -4043\end{array}\right]\)

\(A=\left[\begin{array}{ll}3&-4\\ 1&-1\end{array}\right]=\begin{bmatrix}1&0\\0&1\end{bmatrix}+\begin{bmatrix}2&-4\\1&-2\end{bmatrix}\) (A = I + B)

$B^2=\begin{bmatrix}0&0\\0&0\end{bmatrix}=0⇒B^3=0,B^4=0....B^n=0$ for $n≥2$

so $A^{2022}=[I+B]^{2022}$

$=I^{2022}+2022B+0...$

$=\begin{bmatrix}1&0\\0&1\end{bmatrix}+\begin{bmatrix}4044&-8088\\2022&-4044\end{bmatrix}=\left[\begin{array}{ll}4045 &-8088 \\ 2022 & -4043\end{array}\right]$