  The matrix semigroup membership problem asks, given square matrices
$M,M_1,\ldots,M_k$ of the same dimension, whether $M$ lies in the semigroup
generated by $M_1,\ldots,M_k$. It is classical that this problem is undecidable
in general but decidable in case $M_1,\ldots,M_k$ commute. In this paper we
consider the problem of whether, given $M_1,\ldots,M_k$, the semigroup
generated by $M_1,\ldots,M_k$ contains a non-negative matrix. We show that in
case $M_1,\ldots,M_k$ 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 result is a procedure to
determine, given a matrix $M$, whether the sequence of matrices $(M^n)_{n\geq
0}$ is ultimately nonnegative. This answers a problem posed by S. Akshay
(arXiv:2205.09190). The latter result is in stark contrast to the notorious
fact that it is not known how to determine effectively whether for any specific
matrix index $(i,j)$ the sequence $(M^n)_{i,j}$ is ultimately nonnegative
(which is a formulation of the Ultimate Positivity Problem for linear
recurrence sequences).
