The low-rank matrix recovery problem consists of reconstructing an unknown low-rank matrix from a few linear measurements, possibly corrupted by noise. One of the most popular method in low-rank matrix recovery is based on nuclear-norm minimization, which seeks to simultaneously estimate the most significant singular values of the target low-rank matrix by adding a penalizing term on its nuclear norm. In this paper, we introduce a new method that requires substantially fewer measurements needed for exact matrix recovery compared to nuclear norm minimization. The proposed optimization program utilizes a sparsity promoting regularization in the form of the entropy function of the singular values. Numerical experiments on synthetic and real data demonstrates that the proposed method outperforms stage-of-the-art nuclear norm minimization algorithms.