ABSTRACT
Determinants represent a fundamental mathematical invariant that has been pivotal in the development of modern mathematics. This chapter provides a review of earlier work using many of the tools that are developed as motivation for determinantal results. It deals with the most important properties of 2-by-2 determinants and uses a simple recursive definition for the general case. Sometimes it is possible to give a simple-looking formula for a complicated problem using determinants. These formulas can be quite elegant and can often be proven without direct computation. Because integration has to do with area, it is no surprise that determinants must arise and that dxdt is replaced by the Jacobian determinant. The chapter utilizes row reduction to compute determinants by keeping track of how each row operation changes the determinant. In computing determinants by reduction, the block triangular matrix determinant formula can be used.
