Since the adjacency matrix $\mathrm A$ is not symmetric, we have a directed graph.
Given a positive integer $k$, the $(i,j)$-th entry of $\mathrm A^k$ gives us the number of directed walks of length $k$ from $i$ to $j$. Given the cycle $1 \to 2 \to 3 \to 1$, the entry $(\mathrm A^k)_{11}$ should be $1$ when $k$ is a multiple of $3$ and $0$ when $k$ is not a multiple of $3$. Using SymPy, we can verify this:
>>> A = Matrix([[0,1,0,0,0], [0,0,1,0,0], [1,0,0,1,1], [0,0,0,0,0], [0,0,0,0,0]])>>> A**2[0 0 1 0 0][ ][1 0 0 1 1][ ][0 1 0 0 0][ ][0 0 0 0 0][ ][0 0 0 0 0]>>> A**3[1 0 0 1 1][ ][0 1 0 0 0][ ][0 0 1 0 0][ ][0 0 0 0 0][ ][0 0 0 0 0]>>> A**4[0 1 0 0 0][ ][0 0 1 0 0][ ][1 0 0 1 1][ ][0 0 0 0 0][ ][0 0 0 0 0]>>> A**5[0 0 1 0 0][ ][1 0 0 1 1][ ][0 1 0 0 0][ ][0 0 0 0 0][ ][0 0 0 0 0]>>> A**6[1 0 0 1 1][ ][0 1 0 0 0][ ][0 0 1 0 0][ ][0 0 0 0 0][ ][0 0 0 0 0]Note that $9999$ is a multiple of $3$, whereas $33334$ is not.