Matrices often appear in formal models of social and economic behavior, especially models involving networks. Such models are used to study subjects ranging from opinion dynamics to pollution-mitigation negotiations to the regulation of large marketplace platforms. Matrices are used to capture the focal economic structure in each case. Spectral theory offers powerful tools for understanding matrices, and economic modelers have leveraged these tools to gain considerable insight. When special structure is present, such as nonnegativity or symmetry, more refined tools suited to this structure -- such as Perron--Frobenius theory and the spectral theorem -- offer additional leverage. This essay uses these unifying mathematical threads to offer an accessible tour of several important ideas in social science, assuming minimal non-mathematical background knowledge. Though the introductions to each topic are necessarily brief, the tour cites references throughout for more context.