The Lyapunov matrix and the Lyapunov-Krasovskii functionals of complete type based on the matrix are used to analyze the stability and robust stability of systems with time delays. For systems with constant delay and periodic coefficients, this theory is still at an early stage. In this paper, a new formula for the Lyapunov matrix is presented, which guarantees the existence of the Lyapunov matrix for systems of the considered class and can be easily generalized to wider classes of systems. It is also shown how the new formula can be useful for deriving an exponential stability criterion based on the Lyapunov matrix, similar to that obtained for LTI systems previously.