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...............
 A: Simplify your problem as follows. The Jordan Normal form of your matrix $A$ is 
$$J = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1& 1 \\ 0& 0& 1 \end{pmatrix},$$
where the matrix $P$ is such that $P^{-1}AP = J$ is given by 
$$P = \begin{pmatrix} 0 & 0& 2 \\ -1 & 1 & 1 \\ 1 & 1 &0 \end{pmatrix}.$$
Then $A^{50} = PJ^{50}P^{-1}$. Now what is $J^{50}$?
Edit: Here's how you compute the Jordan Normal form of $A$ and the matrix $P$. First note that it has eigenvalues -1 and 1(with multiplicity 2). The Jordan normal form of $A$ is not diagonal because $A$ is not diagonalisable. It has eigenvectors $(0,1,-1)$ and $(0,1,1)$.
Now by my answer here we already know what the Jordan Normal Form of $A$ looks like, i.e. we know the matrix $J$. How do we obtain the matrix $P$? We now write $J$ like this:
$$J = \left(\begin{array}{c|cc} -1 & 0 & 0 \\ \hline 0 & 1& 1 \\ 0& 0& 1 \end{array}\right).$$
I have put in the grid to emphasize the fact that the first column of the matrix $P$ has to be the eigenvector $(0,-1,1)$ associated to the eigenvalue $-1$. The second column of $P$ looking at the second column of $J$ must now be the eigenvector associated to the eigenvalue $1$. What about the third column of $P$? This is of course given by a basis element for the kernel of $(A - I)^2$. We find that 
$$(A - I)^2 = \begin{pmatrix} 0 & 0 & 0 \\ -1 & 2 & -2 \\ 1 & -2 & 2 \end{pmatrix} $$
which has a basis for its kernel the vector $(2,1,0)$ and so this completes the computation of $P$. 
A: 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. 
A: 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.
