  This paper treats linear algebra over a semiring pair, in a wide range of
applications to tropical algebra and related areas such as hyperrings and fuzzy
rings. First we present a more general category of ``pairs'' with their
morphisms, called ``weak morphisms,'' paying special attention to supertropical
pairs, hyperpairs, and the doubling functor. Then we turn to matrices and the
question of whether the row rank, column rank, and submatrix rank of a matrix
are equal. Often the submatrix rank is less than or equal to the row rank and
the column rank, but there is a counterexample to equality, discovered some
time ago by the second author, which we provide in a more general setting
(``pairs of the second kind'') that includes the hyperfield of signs.
Additional positive results include a version of Cramer's rule, and we find
situations when equality holds, encompassing results by Akian, Gaubert,
Guterman, Izhakian, Knebusch, and Rowen. We pay special attention to the
question of whether $n+1$ vectors of length $n$ need be dependent. At the end,
we introduce a category with stronger morphisms, that preserve a surpassing
relation.
