First one
$$\begin{pmatrix} 2& 3\\5 & 1 \end{pmatrix}^{20}$$
Second one $$A=\begin{pmatrix} 4&0& 0\\0 & 3&0\\2 &0&2 \end{pmatrix}^{20}$$
$$P=\begin{pmatrix} 1&0& 0\\0 & 1&0\\1 &0&1 \end{pmatrix}$$
$$P^{-1}=\begin{pmatrix} 1&0& 0\\0 & 1&0\\-1 &0&1 \end{pmatrix}$$
I have some difficult to solve this problem. For the first one, the characteristic equation is $(\lambda-1)(\lambda-2)-15=\lambda^2-3\lambda-13=0$ that $\lambda$ is $\frac{3\pm\sqrt{61}}2$ is it right?
For the second one, I find $P$ and $P^{-1}$, but multiplication of $P^{-1}AP$ is not diagonal. Is there any case that $A$ cannot be represented as a diagonal but power of $A$ can be calculated?
$A$ has distinct eigenvalues, it must be diagonalizable.
\begin{align}\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}\begin{bmatrix}4 & 0 & 0 \\ 0 & 3 & 0 \\ 2 & 0 & 2 \end{bmatrix} \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} &= \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}\begin{bmatrix}4 & 0 & 0 \\ 0 & 3 & 0 \\ 4 & 0 & 2 \end{bmatrix} \\ &=\begin{bmatrix}4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix} \end{align}