Decomposition problems of boolean matrices are considered, and some interesting results are obtained. We decompose a given boolean matrix into a product of two boolean matrices. The decomposition operation is performed by means of transitivity of a matrix obtained from the given matrix. Decompositions of boolean matrices are important in many applications such as information retrieval, relational databases, large-scale systems, and so on. Boolean matrices represent relations, digraphs, and various binary systems. They are applied to many areas, so that decompositions of boolean matrices play an important role in the areas.