5.3.1. Definition:  Determinant Function

Let A=( a ij )  be an n ´ n matrix with a ij R  or  C .  We denote its ith row by

        A i =( a i1 , a i2 ,, a in )   so that   A=( A 1 A n )

The determinant function of order n is defined as

        d: M n,n R  or  C            by      AdetA=d( A 1 ,, A n )

such that the following axioms are satisfied:

Axiom 1.       Homogeneity In Each Row:

d( ,t A i , )=td( , A i , )            where tR  or  C .

Axiom 2.       Invariance Under Row Addition:

d( , A i + A j ,, A j , )=d( , A i ,, A j , )

Axiom 3.       Scale:

detI=d( I 1 ,, I n )=1

where I k  is the kth unit coordinate vector.