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Determinants of Block Matrices and Schur ’ s Formula by

Introduction. Most undergraduate students become acquainted with the determinant of a matrix by the end of their second or third year of studies. Determinants are usually introduced to classify systems of linear equations. But as many students later learn, determinants are also a rich source of inspiration about algebra, geometry and combinatorics. The purpose of this article is twofold. Our first goal is to discuss Laplace expansions. This important generalization of the standard definition of determinant has quietly slipped away from college textbooks and courses. Laplace expansions lead naturally to questions about the determinant of block matrices. One important consequence is a special case of Schur’s Formula (see Proposition 1). Our second purpose is to present a generalization of Schur’s Formula that is very natural and yet seems to be absent from the vast literature of matrix theory. It is a pleasure to thank our colleague Professor Dan Flath for helpful suggestions.