Collaborative filtering is one of the most popular techniques in designing recommendation systems, and its most representative model, matrix factorization, has been wildly used by researchers and the industry. However, this model suffers from the lack of interpretability and the item cold-start problem, which limit its reliability and practicability. In this paper, we propose an interpretable recommendation model called Multi-Matrix Factorization (MMF), which addresses these two limitations and achieves the state-of-the-art prediction accuracy by exploiting common attributes that are present in different items. In the model, predicted item ratings are regarded as weighted aggregations of attribute ratings generated by the inner product of the user latent vectors and the attribute latent vectors. MMF provides more fine grained analyses than matrix factorization in the following ways: attribute ratings with weights allow the understanding of how much each attribute contributes to the recommendation and hence provide interpretability; the common attributes can act as a link between existing and new items, which solves the item cold-start problem when no rating exists on an item. We evaluate the interpretability of MMF comprehensively, and conduct extensive experiments on real datasets to show that MMF outperforms state-of-the-art baselines in terms of accuracy.