The class of fully copositive (Cf0) matrices is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices. It is shown that fully copositive matrices within the class of Q0-matrices are P0-matrices. As a corollary of this main result, we establish that a bisymmetric Q0-matrix is positive semidefinite if, and only if, it is fully copositive. Another important result of the paper is a constructive characterization of Q0-matrices within the class of Cf0. While establishing this characterization, it will be shown that Graves's principal pivoting method of solving Linear Complementarity Problems (LCPs) with positive semidefinite matrices is also applicable to Cf0 ∩ Q0 class. As a byproduct of this characterization, we observe that a Cf0-matrix is in Q0 if, and only if, it is completely Q0. Also, from Aganagic and Cottle's result, it is observed that LCPs arising from Cf0 ∩ Q0 class can be processed by Lemke's algorithm.
