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In this work, we propose a novel representation of complex multi-relational networks, which is compact and allows very efficient network analysis. Multi-relational networks capture complex data relationships and have a variety of applications, ranging from biomedical to financial, social, etc. As they get to be used with ever larger quantities of data, it is crucial to find efficient ways to represent and analyse such networks. This paper introduces the concept of Prime Adjacency Matrices (PAMs), which utilize prime numbers, to represent the relations of the network. Due to the fundamental theorem of arithmetic, this allows for a lossless, compact representation of a complete multi-relational graph, using a single adjacency matrix. Moreover, this representation enables the fast computation of multi-hop adjacency matrices, which can be useful for a variety of downstream tasks. We illustrate the benefits of using the proposed approach through various simple and complex network analysis tasks.