It was proved in Chapter 2 that the scalar triple product of 3 vectors
can be written as
which is non-zero iff the row vectors are
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
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.