In cases where we do not have the exact parameter values of a mathematical model, we often have at least some structural information, e.g., that some parameters are nonzero. Such information can be captured by so-called pattern matrices, whose symbolic entries are used to represent the available information about the corresponding parameters. In this letter, we focus on pattern matrices with three types of symbolic entries: those that represent zero, nonzero, and arbitrary parameters. We formally define and study addition and multiplication of such pattern matrices. The results are then used in the algebraic characterization of three strong structural properties. In particular, we provide sufficient conditions for controllability of linear descriptor systems, necessary and sufficient conditions for input-state observability, and sufficient conditions for output controllability of linear systems.