  We consider the following matrix reachability problem: given $r$ square
matrices with entries in a semiring, is there a product of these matrices which
attains a prescribed matrix? We define similarly the vector (resp. scalar)
reachability problem, by requiring that the matrix product, acting by right
multiplication on a prescribed row vector, gives another prescribed row vector
(resp. when multiplied at left and right by prescribed row and column vectors,
gives a prescribed scalar). We show that over any semiring, scalar reachability
reduces to vector reachability which is equivalent to matrix reachability, and
that for any of these problems, the specialization to any $r\geq 2$ is
equivalent to the specialization to $r=2$. As an application of this result and
of a theorem of Krob, we show that when $r=2$, the vector and matrix
reachability problems are undecidable over the max-plus semiring
$(Z\cup\{-\infty\},\max,+)$. We also show that the matrix, vector, and scalar
reachability problems are decidable over semirings whose elements are
``positive'', like the tropical semiring $(N\cup\{+\infty\},\min,+)$.
