## 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 *A*_{1} *A*_{2}
and *A*_{3} [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.