The class of matrices Q0, characterized as all n by n matrices M, for which the linear complementarity problem, y = Mx + q, y ⩾ 0, x ⩾ 0, ytx = 0 has a solution whenever y = Mx + q, y ⩾ 0, x ⩾ 0 has a solution, is embedded in a class 𝒬0 of n by 2n matrices A satisfying: if q ∈ Pos(A) = {Au: u ⩾ 0}, then Aw = q has a complementary solution wt = (yt, xt) ⩾ 0 with ytx = 0. Here y and x are n-vectors. A characterization of 𝒬0 is established. As a consequence of this, a new characterization of the matrix class 𝒬0 as well as some other relevant results are obtained.
