The Determinant

Rather than using the traditional approach via cofactor expansion, determinants are introduced as a function defined on square matrices having properties D1-D3, where D1 is normalization and D2-D3 describe the impact on the determinant of two row operations. This approach allows students to begin computing determinants using Gauss-Jordan elimination with the associated software. Properties of determinants of elementary matrices are derived, and these properties are used to prove the important theorem that the determinant of a product of matrices is the product of their determinants. Another application of elementary matrices is the theorem that the determinant is the unique function D with properties D1-D3. Uniqueness is emphasized as an important theme in mathematics, and this uniqueness is later used to prove important properties of the determinant. The process of cofactor expansion is introduced, but the proof that cofactor expansion actually computes the determinant is deferred until a later chapter. The fact that cofactor expansion is an extremely inefficient method to compute determinants is supported by describing the factorial growth rate of the method. Students may practice cofactor expansion of 3x3 matrices in the associated software.