In this paper, we address the estimation problem for the solution of the discrete algebraic matrix Riccati equation. Both upper and lower bounds are measured. Compared to the majority of the approaches proposed in the literature, the present results are sharper. We also apply the results obtained to solve the robust stabilization problem of discrete time-delay systems. A robust stabilizability criterion and the corresponding state feedback control law are proposed. Furthermore, the tolerable bound of the delay term is also estimated. Finally, numerical examples are given to demonstrate the applications of the results.