A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A, q) has a solution for all q ∈ Rn. This means that for every vector q there exists a vector x such that x ≥ 0, y = Ax + q ≥ 0, and xT y = 0. A well-known result of Karamardian states that if the problems LCP(A, 0) and LCP(A, d) for some d ∈ Rn; d > 0 have only the zero solution, then A is a Q-matrix. Upon relaxing the requirement on the vectors d and y so that the vector y belongs to the translation of the nonnegative orthant by the null space of AT, d belongs to its interior, and imposing the additional condition on the solution vector x to be in the intersection of the range space of A with the nonnegative orthant, in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, wherein these two modified problems have only zero as a solution. In this article, a systematic treatment of these matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of Q-matrices. A subclass of a recently introduced notion of P#-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of P#-matrices and make some fundamental contributions. © 2021, International Linear Algebra Society. All rights reserved.