<s>[INST] for a $3 \times 3$ matrix A ,value of $ A^{50} $ is I f
$$A= \begin{pmatrix}1& 0 & 0 \\ 
1 & 0 & 1\\
0 & 1 & 0 \end{pmatrix}$$
 then $ A^{50} $  is


*

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

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

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

*$$\begin{pmatrix}1& 0 & 0 \\ 
24 & 1 & 0\\
24 & 0 & 1\end{pmatrix}$$
I am stuck on this problem. Can anyone help me please...............
 [/INST] The answer is 3. $$\begin{pmatrix}1& 0 & 0 \\ 
25 & 1 & 0\\
25 & 0 & 1 \end{pmatrix}$$
Just compute $A^2$ , $A^3$ , $A^4$ and $A^5$ and you will understand the repeated pattern. 
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