If H is a nxn primitive, irreducible matrix, is it always true that Hn-1 > 0? That is, every entry in Hn-1 is positive. From my class notes, the definition of H primitive is that there exists some k>0 such that Hk > 0. And a matrix is irreducible if its digraph is strongly connected (that is, it is possible to reach every node from every other node in a finite number of steps). I am interpreting H as a matrix of a strongly connected digraph. Since it is strongly connected, it should be possible to reach any other node after n-1 steps, and thus (H n -1)ij should be positive since there is a positive probability that it will get to node j from node i in n-1 steps. Is there a good way to prove that every element of Hn-1 must be positive?