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 $P$.