Even though subgraph counting and subgraph matching are well-known NP-Hard problems, they are foundational building blocks for many scientific and commercial applications. In order to analyze graphs that contain millions to billions of edges, distributed systems can provide computational scalability through search parallelization. One recent approach for exposing graph algorithm parallelization is through a linear algebra formulation and the use of the matrix multiply operation, which conceptually is equivalent to a massively parallel graph traversal. This approach has several benefits, including 1) a mathematically-rigorous foundation, and 2) ability to leverage specialized linear algebra accelerators and high-performance libraries. In this paper, we explore and define a linear algebra methodology for performing exact subgraph counting and matching for 4-vertex subgraphs excluding the clique. Matches on these simple subgraphs can be joined as components for a larger subgraph. With thorough analysis, we demonstrate that the linear algebra formulation leverages path aggregation which allows it to be up 2x to 5x more efficient in traversing the search space and compressing the results as compared to tree-based subgraph matching techniques.