We describe the utility of matrices in the solution of linear systems of differential equations with constant coefficients, via the matrix exponential. This can be comprehended by readers who have not taken a course in differential equations, thanks to the analogy with the simple equation of radioactive decay. We delineate the properties of the matrix exponential and briefly survey the issues involved in its computation. The interesting question here (in theory, at least) is the exponential of a defective matrix. Although we direct readers elsewhere for a rigorous proof of the Jordan decomposition theorem, we work out the format of the resulting exponential. Many authors ignore, mislead, or confuse their readers in the calculation of the generalized eigenvector Jordan chains of a defective matrix, so we describe a straightforward and foolproof procedure for this task. The alternative calculation of the matrix exponential, based on the primary decomposition theorem and forgoing the Jordan chains, is also presented. Projects address positive definite matrices, Hessenberg forms, the discrete Fourier transform, advanced aspects of the singular value decomposition, and matrix deflation.