Matrix models are often used to predict the dynamics of size-structured or age-structured populations. The asymptotic behaviour of such models is defined by their malthusian growth rate [lambda], and by their stationary distribution w that gives the asymptotic proportion of individuals in each stage. As the coefficients of the transition matrix are estimated from a sample of observations, [lambda] and w can be considered as random variables whose law depends on the distribution of the observations. The goal of this study is to specify the asymptotic law of [lambda] and w when using the maximum likelihood estimators of the coefficients of the transition matrix. We prove that [lambda] and w are asymptotically normal, and the expressions of the asymptotic variance of [lambda] and of the asymptotic covariance matrix of w are given. The convergence speed of [lambda] and w towards their asymptotic law is studied using simulations. The results are applied to a real case study that consists of a Usher model for a tropical rain forest in French Guiana. They permit to assess the number of trees to measure to get a given precision on the estimated asymptotic diameter distribution, which is an important information on tropical forest management.