Regular Path Queries (RPQs), which are essentially regular expressions to be matched against the labels of paths in labeled graphs, are at the core of graph database query languages like SPARQL. A way to solve RPQs is to translate them into a sequence of operations on the adjacency matrices of each label. We design and implement a Boolean algebra on sparse matrix representations and, as an application, use them to handle RPQs. Our baseline representation uses the same space as the previously most compact index for RPQs and outperforms it on the hardest types of queries -- those where both RPQ endpoints are unspecified. Our more succinct structure, based on $k^2$-trees, is 4 times smaller than any existing representation that handles RPQs, and still solves complex RPQs in a few seconds. Our new sparse-matrix-based representations dominate a good portion of the space/time tradeoff map, being outperformed only by representations that use much more space. They are also of independent interest beyond solving RPQs.