Supplemental Materials for EE203001 Students II. Determinant Functions

1 Three Axioms for a Determinant Function Let d be a scalar-valued function on the space Mn×n of all n×n matrices. Let A1, A2, . . . , An be the n rows of an n×n matrix A. We will denote the value d(A) of A under the function d as d(A) = d(A1, A2, . . . , An), indicating that d is a scalar-valued function of n n-dimensional row vectors A1, A2, . . . , An. The scalar-valued function d is called a determinant function of order n if it satisfies the following three axioms: Axiom 1. Homogeneity in each row . If matrix B is obtained from matrix A by multiplying one row, says the ith row, of A by a scalar α, then d(B) = α d(A), i.e., d(A1, . . . , αAi, . . . , An) = α d(A1, . . . , Ai, . . . , An).