The matrix semigroup membership problem asks, given square matrices M,M₁,…,M_k of the same dimension, whether M lies in the semigroup generated by M₁,…,M_k. It is classical that this problem is undecidable in general, but decidable in case M₁,…,M_k commute. In this paper we consider the problem of whether, given M₁,…,M_k, the semigroup generated by M₁,…,M_k contains a non-negative matrix. We show that in case M₁,…,M_k commute, this problem is decidable subject to Schanuel’s Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability proof is a procedure to determine, given a matrix M, whether the sequence of matrices (Mⁿ)_{n = 0}^∞ is ultimately nonnegative. This answers a problem posed by S. Akshay [S. Akshay et al., 2022]. The latter result is in stark contrast to the notorious fact that it is not known how to determine, for any specific matrix index (i,j), whether the sequence (Mⁿ)_{i,j} is ultimately nonnegative. Indeed the latter is equivalent to the Ultimate Positivity Problem for linear recurrence sequences, a longstanding open problem.