There has been a significant interest in the recovery of low-rank matrices from an incomplete of measurements, due to both theoretical and practical developments demonstrating the wide applicability of the problem. A number of methods have been developed for this recovery problem, however, a principled method for choosing the unknown target rank is generally missing. In this paper, we present a recovery algorithm based on sparse Bayesian learning (SBL) and automatic relevance determination principles. Starting from a matrix factorization formulation and enforcing the low-rank constraint in the estimates as a sparsity constraint, we develop an approach that is very effective in determining the correct rank while providing high recovery performance. We provide empirical results and comparisons with current state-of-the-art methods that illustrate the potential of this approach.