We establish a set of new sufficient conditions for the Hurwitz and Schur stability of interval matrices. We use these results to establish necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices. We relate the above results to the existence of quadratic Lyapunov functions for linear time-invariant systems with interval-valued coefficient matrices. Using the above results, we develop an algorithm to determine the Hurwitz and the Schur stability properties of interval matrices. We demonstrate the applicability of our results by means of two specific examples.