Abstract We investigate real matrices A such that each matrix with the same sign pattern as A has a nonzero, nonnegative vector in its nullspace. For geometric reasons we call these matrices sign-central. These matrices were introduced and given a combinatorial characterization by Davydov and Davydova. We give an alternative proof of their characterization. We show that under a minimality assumption a sign-central matrix with m nonzero rows has at least m + 1 columns, and that equality holds if and only if the matrix is an S-matrix (as defined in the theory of sign-solvability).