For nonsmooth systems with dry friction, the transitional matrix was deduced from the cocycle at the nonsmooth instants, and then a numerical method for calculating Lyapunov exponents was introduced by means of the transitional matrix. The method was conducted from conventional algorithm of calculating Lyapunov exponents for smooth systems. The dynamical behavior of a system with dry friction was analyzed by calculating its Lyapunov exponents, along with time history and Poincare map; and numerical results showed the effectiveness of the method. Compared with some existent methods of calculating Lyapunov exponents of nonsmooth systems, the method is more adequate to numerical calculation.