<s>[INST] Find $A^{50}$ for a $3\times 3$ matrix 
If $$A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$$ then $A^{50}$ is:

*

*$\begin{bmatrix}1&0&0\\25&1&0\\25&0&1\end{bmatrix}$


*$\begin{bmatrix}1&0&0\\50&1&0\\50&0&1\end{bmatrix}$


*$\begin{bmatrix}1&0&0\\48&1&0\\48&0&1\end{bmatrix}$


*$\begin{bmatrix}1&0&0\\24&1&0\\24&0&1\end{bmatrix}$

For $A$, the eigenvalues are $1, 1, -1$, but I don't know the procedure any further.
 [/INST] Sometimes a good way to start a problem like this is just to try a few multiplications and look for a pattern.
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)$ 
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)$$=$
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & 0 & 1 \end{array} \right)$ $=A^2$
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & 0 & 1 \end{array} \right)$ 
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)$$=$
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
2 & 0 & 1 \\
1 & 1 & 0 \end{array} \right)$ $=A^3$
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
2 & 0 & 1 \\
1 & 1 & 0 \end{array} \right)$ 
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)$$=$
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
2 & 1 & 0 \\
2 & 0 & 1 \end{array} \right)$ $=A^4$
Now the pattern should be clear enough to be able to solve the multiple-choice question.  If not, do two more.
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