[For part I see Linear Multilinear Algebra 38, 241-247 (1995; Zbl 0826.15017)]. The basic goal is to determine when a \(Z\)-matrix \(A\) (\(A=sI-P\) for \(P\geq 0\)) is an irreducible \(M\)-matrix (so \(s>\rho(P)\), the spectral radius of \(P\)). The cases for \(A\) singular and nonsingular are treated separately, and in each case a few necessary and sufficient conditions are given. The main reference in this area is the book of A. Berman and R. J. Plemmons [Nonnegative matrices in the mathematical sciences (1979; Zbl 0484.15016)].