We present a fast, simple matrix method of computing the unique invariant measure and associated Lyapunov exponents of a nonlinear iterated function system. Analytic bounds for the error in our approximate invariant measure (in terms of the Hutchinson metric) are provided, while convergence of the Lyapunov exponent estimates to the true value is assured. As a special case, we are able to rigorously estimate the Lyapunov exponents of an iid random matrix product. Computation of the Lyapunov exponents is carried out by evaluating an integral with respect to the unique invariant measure, rather than following a random orbit. For low dimensional systems, our method is considerably quicker and more accurate than conventional methods of exponent computation. An application to Markov random matrix product is also described.