THE GEOMETRY OF DETERMINANTS

The determinant is a “machine” which “processes” the column (or row) vectors of a square matrix A, producing a number denoted by det A. As a function of the column vectors of A, det is (1) linear, (2) skew-symmetric, and (3) normalized. The purpose of this paper is to show how these three properties, which completely characterize the determinant, can be deduced from a geometric definition of the determinant as a “calculator of areas and volumes.” I believe that this approach has a pedagogical advantage over the traditional algebraic definition because many of the familiar properties of the determinant are more easily conceptualized geometrically than algebraically.