For A ∈ ℝn×n and q ∈ ℝn, the linear complementarity problem LCP(A; q) is to determine if there is x ∈ ℝn such that x ≥ 0; y = Ax+q ≥ 0 and xT y = 0. Such an x is called a solution of LCP(A; q). A is called an R 0-matrix if LCP(A; 0) has zero as the only solution. In this article, the class of R0-matrices is extended to include typically singular matrices, by requiring in addition that the solution x above belongs to a subspace of ℝn. This idea is then extended to semidefinite linear complementarity problems, where a characterization is pre-sented for the multplicative transformation.