Invertibility of NxN nonnegative matrix with diagonally dominant elements

Question

I have a NxN nonnegative matrix where the diagonal element of any row i is greater than the off diagonal elements, 1 > aii > aij ≥ 0 for j not equal to i . This not a diagonally dominant matrix as it is possible that aii < ∑aij over j not equal to i. I want to know if this matrix is generically invertible for arbitrary (random) coefficients maintaining the restriction, 1 > aii > aij ≥ 0. If the matrix is generically invertible a reference would be most helpful.