On co-positive, semi-monotone Q-matrices

In this paper we consider not necessarily symmetric co-positive as well as semi-monotone Q-matrices and give a set of sufficient conditions for such matrices to be R0-matrices. We give several examples to show the sharpness of our results. Construction of these examples is based on the following elementary proposition: If A is a square matrix of ordern whose first two rows are identical and both A11 and A22 are Q-matrices where Aii stands for the principal submatrix of A obtained by deleting row i and column i from A, then A is a Q-matrix