I need to calculate $x^{50}$ $x=\begin{pmatrix}1&0&0\\1&0&1\\0&1&0\end{pmatrix}$, I need to calculate $x^{50}$
Could anyone tell me how to proceed?
Thank you.

The Jordan Decomposition yields
$$
\left[
\begin{array}{r}
 1 & 0 & 0 \\
 1 & 0 & 1 \\
 0 & 1 & 0
\end{array}
\right]
=
\left[
\begin{array}{r}
 0 & 0 & 2 \\
 -1 & 1 & 1 \\
 1 & 1 & 0
\end{array}
\right]
\left[
\begin{array}{r}
 -1 & 0 & 0 \\
 0 & 1 & 1 \\
 0 & 0 & 1
\end{array}
\right]
\left[
\begin{array}{r}
 0 & 0 & 2 \\
 -1 & 1 & 1 \\
 1 & 1 & 0
\end{array}
\right]^{-1}
$$
Block matrices are easier to raise to a power:
$$
\begin{align}
\left[
\begin{array}{r}
 1 & 0 & 0 \\
 1 & 0 & 1 \\
 0 & 1 & 0
\end{array}
\right]^{50}
&=
\left[
\begin{array}{r}
 0 & 0 & 2 \\
 -1 & 1 & 1 \\
 1 & 1 & 0
\end{array}
\right]
\left[
\begin{array}{r}
 -1 & 0 & 0 \\
 0 & 1 & 1 \\
 0 & 0 & 1
\end{array}
\right]^{50}
\left[
\begin{array}{r}
 0 & 0 & 2 \\
 -1 & 1 & 1 \\
 1 & 1 & 0
\end{array}
\right]^{-1}\\[6pt]
&=
\left[
\begin{array}{r}
 0 & 0 & 2 \\
 -1 & 1 & 1 \\
 1 & 1 & 0
\end{array}
\right]
\left[
\begin{array}{r}
 1 & 0 & 0 \\
 0 & 1 & 50 \\
 0 & 0 & 1
\end{array}
\right]
\left[
\begin{array}{r}
 0 & 0 & 2 \\
 -1 & 1 & 1 \\
 1 & 1 & 0
\end{array}
\right]^{-1}\\[6pt]
&=
\left[
\begin{array}{r}
 1 & 0 & 0 \\
 25 & 1 & 0 \\
 25 & 0 & 1
\end{array}
\right]
\end{align}
$$
