Inverses of M-type matrices created with irreducible eventually nonnegative matrices

An M-matrix is a matrix that can be expressed as αI-P, where P is entry wise nonnegative and α⩾ρ(P). It is well known that the inverse of a nonsingular irreducible M-matrix is positive. In this paper, matrices of the form αI-P, where P is an irreducible eventually nonnegative matrix and α>ρ(P), are studied. It is shown that if index0(P)⩽1, then there exists a positive number λ such that if λ>α>ρ(P), then the inverse of αI-P is positive.