Numerical Linear Algebra with Applications

Chapter 4

Determinants

Abstract

This chapter introduces the determinant. It begins by giving the definition of the determinant in terms of sums of permutations, an unwieldy way to compute it. This definition provides a formula for the determinant of a 2 × 2 and a 3 × 3 matrix, and these motivate expansion by minors. Expansion by minors is presented in general form and used to provide a simple means of computing the determinant of a lower and upper triangular matrix and a diagonal matrix. After presenting the adjoint, the chapter shows how to compute the inverse using the adjoint. Cramer’s rule is presented, and it is made clear it should be used only for theoretical purposes for matrices larger than 3 × 3. The chapter shows how the determinant ...