Matrix bounds, upper and lower, for the solution of the discrete algebraic Riccati and Lyapunov equations respectively, are proposed in this paper. They are new or sharper than the majority of existing results. By making use of these new matrix bounds, the corresponding bounds for each eigenvalue, the trace and the determinant of these solution matrices are also presented. Comparisons are made between these bounds and the majority of those appearing in the literature. It is shown that the obtained results are tighter. Finally, we give illustrative examples to show that these estimates appear to be considerably stronger than previously available results in many cases.