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...............

You should learn BenjaLim's answer, which provides a general method for dealing with this kind of problems. However, here is a simple answer just for fun. Note that
$$
A^2= 
\begin{pmatrix}
1&0&0\\ 
1&1&0\\
1&0&1\end{pmatrix}
=I+\underbrace{\begin{pmatrix}
0&0&0\\ 
1&0&0\\
1&0&0\end{pmatrix}}_{L}
$$
and $L^2=0$. Therefore
$$
A^{50} = (I+L)^{25} = I+{25\choose 1}L+\sum_{k=2}^{25}{25\choose k}L^k=I+25L
$$
and hence the answer is 3.
