Matrix irreducibility. Strongly connected graph

Let $$A = \left[\begin{array}{ccc} 0 &1& 0\\ 0 &0 &1\\ 1 &0& 0 \end{array}\right],$$ then $$ A^1 = \left[\begin{array}{ccc} 0 &1& 0\\ 0 &0 &1\\ 1 &0& 0 \end{array}\right]\quad A^2 = \left[\begin{array}{ccc} 0 &0& 1\\ 1 &0 &0\\ 0 &1& 0 \end{array}\right]\quad A^3 = \left[\begin{array}{ccc} 1 &0& 0\\ 0 &1 &0\\ 0 &0& 1 \end{array}\right], $$ so for every pair $\langle i,j \rangle$ there is a power $m$ such that $(A^m)_{i,j} > 0$ and the matrix is irreducible. However, this is not a magical statement, worded in graph-theoretic form we have

for every pair $\langle i,j \rangle$ there is a length $m$ such that there is a directed path of length $m$ between $i$ and $j$

and this is true only if the graph is strongly connected. Moreover, take a reducible matrix $B$:

$$B = \left[\begin{array}{ccc} B_1 &B_2\\ 0 &B_3 \end{array}\right]$$

and consider its corresponding graph. Observe that there is no edge from vertices of block $B_3$ to vertices of block $B_2$. In other words, once you get to $B_2$, there is no way out, in particular the graph cannot be strongly connected.

I hope this helps $\ddot\smile$