5.2. Motivation for the Choice of Axioms for a Determinant Functions

It was proved in Chapter 2 that the scalar triple product of 3 vectors

A 1 =( a 11 , a 12 , a 13 )     A 2 =( a 21 , a 22 , a 23 )      A 3 =( a 31 , a 32 , a 33 )

can be written as

        ( A 1 × A 2 ) A 3 =det( A 1 A 2 A 3 )   =det(    a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33    )

which is non-zero iff the row vectors are independent.  Furthermore, it is easily shown geometrically that ( A 1 × A 2 ) A 3  is equal to the volume of the parallelepiped whose sides are A1 A2 and A3 [see Fig.5.1].  If the vectors are dependent, they are coplanar so that the triple product, and hence the volume, vanishes.

 

Later on, we shall define the general determinant function d as an n-D volume.  The 3-D case thus serves as a prototype from which the basic characteristics of such a “volume” function can be deduced.  [ Note: it is more rigorous, as well as simpler, to start with the 2-D case and deal with an “area” function.  Nevertheless, we shall follow Apostol’s approach here].  Consider then the determinant function

        d( A 1 , A 2 , A 3 )=( A 1 × A 2 ) A 3

As a “volume” function, it satisfies the following easily verified geometrical properties:

1.         If a row (side) is multiplied by a factor t, so is the determinant (volume).

2.         If a row (side) is replaced by the sum of itself and another row (vector sum of it and another side), the determinant (volume) is unchanged.

3.         The determinant (volume) of the 3 unit coordinate vectors, which forms the unit cube, is 1.

 

These will serve as the bases for the axiomatic definition of the determinant function to be described in the next section.