Matrix-based representations provide a natural and effective way to encode analytic operations arising in generating-function calculus and polynomial systems. In this work, we explore a finite-dimensional matrix-algebraic framework associated with analytic generating kernels of bivariate s–Appell type. By introducing appropriate Wronskian coordinate vectors, we construct families of structured lower-triangular matrices of the Pascal type that act naturally on polynomial coordinate spaces and reflect underlying analytic transformations. This matrix formulation makes it possible to reinterpret classical analytic operations such as differentiation, translation, and recurrence in terms of matrix actions and binomial-type convolutions. Within this unified setting, several structural properties of the associated polynomial sequences, including recurrence relations, shift identities, and differential-type relations, can be systematically examined from a linear-algebraic and operator-theoretic perspective. The approach allows different classes of bivariate polynomial systems, including classical Appell-type constructions and more general systems, to be treated within a common finite-dimensional framework. By organizing generating-function-based structures through matrix algebras, the proposed viewpoint offers a coherent analytical tool for studying polynomial systems and their associated transformations. Overall, this work highlights the usefulness of structured matrix methods in mathematical analysis and demonstrates how finite-dimensional operator representations can provide insight into the analytic behavior and algebraic structure of bivariate polynomial families.