The determinant

A (mostly) self-contained introduction to the determinant aimed at advanced undergraduates, written for the Math 350H class at Rutgers spring semester of 2017. Most of the material is taken from [1, 2], but it has been substantially reorganized. The goal of this article is to present an organized and motivated discussion of the abstract definition of the determinant and its most basic properties. 1 Notation 2 2 Motivation: signed area and its properties 2 2.1 Signed area of a parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Properties of signed area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Implications of these properties . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 An application for 2× 2 matrices 5 4 Background 6 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Implication of the alternating property . . . . . . . . . . . . . . . . . . . . . 7 4.3 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.4 Permutation groups and alternating n-linear functions . . . . . . . . . . . . . 8 5 Defining the determinant for n× n matrices 10 5.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3 Conclusion: the existence of a unique determinant . . . . . . . . . . . . . . . 14 6 Additional properties of the determinant 14 7 Final thoughts 16 7.1 Interpretation and uses of determinant . . . . . . . . . . . . . . . . . . . . . 16 7.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17