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A real square matrix  A  is called a  Q -matrix if the linear complementarity problem LCP( A ,  q ) has a solution for all. This means that for every vector  q , there exists a vector  x  ≥ 0 such that  y  =  Ax  +  q  ≥ 0 and  xTy  = 0. Two new classes of matrices are studied, namely the  Q# -matrices and-matrices. If for every vector  q  ∈  R ( A ), there exists a vector  x  ∈  R ( A ) satisfying  x  ≥ 0,  y  =  Ax  +  q  ≥ 0 and  xTy  = 0, then  A  is a  Q# -matrix. If the vector  x  satisfying the above properties is instead required to be in  R ( AT), then  A  is a-matrix. Properties of these classes of matrices and their relationship with the class of  Q -matrices are studied.

Q # -matrices and Q † -matrices: two extensions of the Q -matrix concept