TITLE: Proof that a strongly connected digraph has an irreducible adjacency matrix
QUESTION [5 upvotes]: I need to prove that a strongly connected digraph has an irreducible adjacency matrix. If anybody would be willing to give an advice on how to tackle this problem I would be thankful.

REPLY [2 votes]: Hint:
Assume that the matrix is reducible, let $V_1, V_2, \ldots, V_k$ be the vertices that form blocks in the reduced form of matrix (that is, it is upper triangular with respect to those blocks). Show that there is no edge $v_{k+l}\to v_k$ for $v_k \in V_k$ and $v_{k+l} \in V_{k+l}$ by observing that the related number in the matrix is 0 (zero).
Good luck!