In this paper we study the positive solutions of a cooperative system of any number of equations which consider the case of the slow diffusion and include the Lotka-Volterra model. We determine conditions of existence of global solution and blow-up in finite time in terms of the value of the spectral radius of a certain nonnegative matrix associated to the system. The results generalize the ones known for the particular case of two equations and we justify them by using the specific properties of nonnegative matrices which translate the cooperative character of the system.