I would like to aply this theorem in a pratic an exercise, the theorem i: A nonnegative square matrix $A=\left(a_{ij}\right)$ is irreducible if and only if for each $(i, j)$ there exists an integer $к$ such that $a_{ij}^k>0$.
How do i find this integer k? For example, give the matrix $A=\left[\begin{matrix} 0 & 1 & 1\\ 1 & 0 & 0\\ 1 & 0 & 0 \end{matrix}\right]$
But i understand if i have to prove with the property $\left(I+A\right)^{n-1}$, because i know $n-1=3-1=2$
$k$ depends on $i$ and $j$. If you view $A$ as an adjacency matrix for a directed graph $G$, so that $a_{ij}>0$ if and only if there is a directed arc in $G$ joining node $i$ to node $j$, then $(A^k)_{ij}>0$ if and only if there is a path of length $k$ joining node $i$ to node $j$.
In your example, the graph contains the following paths: \begin{aligned} &1\to2\to1,\\ &1\to2,\\ &1\to3,\\ &2\to1,\\ &2\to1\to2,\\ &2\to1\to3,\\ &3\to1,\\ &3\to1\to2,\\ &3\to1\to3. \end{aligned} Therefore, for $(i,j)=(1,1),(2,2),(2,3),(3,2)$ and $(3,3)$, you may take $k=2$. For other $(i,j)$s, you may take $k=1$.