  An $L$-matrix is a matrix whose off-diagonal entries belong to a set $L$, and
whose diagonal is zero. Let $N(r,L)$ be the maximum size of a square $L$-matrix
of rank at most $r$. Many applications of linear algebra in extremal
combinatorics involve a bound on $N(r,L)$. We review some of these
applications, and prove several new results on $N(r,L)$. In particular, we
classify the sets $L$ for which $N(r,L)$ is linear, and show that if $N(r,L)$
is superlinear and $L\subset \mathbb{Z}$, then $N(r,L)$ is at least quadratic.
  As a by-product of the work, we asymptotically determine the maximum
multiplicity of an eigenvalue $\lambda$ in an adjacency matrix of a digraph of
a given size.
