Non negative irreducible matrix times a positive matrix

Question

Let $A$ be a real non negative irreducible matrix, meaning that $(a_{ij})\geq 0$ I know from a theorem in Minc's book that $(I+A)^{n-1}>0$.

It is also stated in the book that $B=(I+A)^{n-1}.A>0$. I am not convinced. How can I see this? Could it be that $B=(I+A)^{n-1}.A\geq 0$?

Thanks in advance.

Answer 1

Denote the $j$-column of $A$ by $a_{\ast j}$. As $A$ is irreducible and non-negative, $a_{\ast j}$ is not the zero vector and $a_{\ast j}\ge0$ entrywise. Since $B>0$, we get $Ba_{\ast j}>0$ for each $j$. Thus $BA>0$.