# 22.17 Eventual non-negativity of Matrices

Consider the problem: given a matrix $$M$$ over integers, does there exist a natural number $$n$$ such that $$M^n$$ has only non-negative entries? Is this question as hard as ultimate positivity? Note that if you consider two matrices $$M, N$$ and ask if there exists $$n$$ such that $$M^n+ N^n$$ has only non-negative entries, that problem is indeed equivalent to ultimate positivity, but with a single matrix, we do not know. Note also that if we ask strict positivity of entries, the single matrix case becomes easy!