In this paper, we develop a new formula for the characteristic equation of a matrix A explicitly and directly in terms of the elements of A. This contrasts with existing methods such as Leverrier's algorithm which is iterative and eigenvalue computation based methods which cannot be applied to matrices with unknown parameters. The formula is applied here to derive explicit stability conditions in terms of the matrix elements for continuous and discrete time systems. It is also shown how the characteristic equation of time-delay systems may be determined. It is expected that the formula will have myriad other uses in control theory and computational linear algebra. Illustrative application examples are included.