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

Abstract An M -matrix is a matrix that can be expressed as α I - 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 α I - P , where P is an irreducible eventually nonnegative matrix and α > ρ ( P ) , are studied. It is shown that if index 0 ( P ) ⩽ 1 , then there exists a positive number λ such that if λ > α > ρ ( P ) , then the inverse of α I - P is positive.