  Given a finite set of matrices with integer entries, the matrix mortality
problem asks if there exists a product of these matrices equal to the zero
matrix. We consider a special case of this problem where all entries of the
matrices are nonnegative. This case is equivalent to the NFA mortality problem,
which, given an NFA, asks for a word $w$ such that the image of every state
under $w$ is the empty set. The size of the alphabet of the NFA is then equal
to the number of matrices in the set. We study the length of shortest such
words depending on the size of the alphabet. We show that for an NFA with $n$
states this length can be at least $2^n - 1$ for an alphabet of size $n$,
$2^{(n - 4)/2}$ for an alphabet of size $3$ and $2^{(n - 2)/3}$ for an alphabet
of size $2$. We also discuss further open problems related to mortality of NFAs
and DFAs.
