Suppose $A$ is an $n\times n$ irreducible nonnegative matrix, prove that $(E+A)^{n-1}$ is a nonnegative matrix with positive elements.
I think that this excise can be proved by definition. So I tried to prove by contradiction. I suppose there exists an element of $(E+A)^{n-1}$ equals zero and then deduce to a contradiction, but I didn't know how, since I needed to consider so many situations.