Linear Algebra, 2020a

334 Chapter Four. Determinants
2.6 Example Determinants bigger than 3×3 go quickly with the Gauss’s Method
procedure.
1 0 1 3
1 0 1 3
1 0 1 3
0 1 1 4
0 1 1 4
0 1 1 4
=
=−
=−(−5) =5
0 0 0 5
0 0 0 5
0 0 −1 −3
∣0 1 0 1∣
∣0 0 −1 −3∣
∣0 0 0 5 ∣
That example raises an important point. This chapter’s introduction gives
formulas for 2×2 and 3×3 determinants, so we know that they exist, but not for
determinant functions on matrices that are 4×4 or larger. Instead, Definition 2.1
gives properties that a determinant function should have and leads to computing
determinants by Gauss’s Method.
However, for any matrix we can reduce it to echelon form by Gauss’s Method
in multiple ways. For example, given a reduction we could change it by inserting
a first step that multiplies the top row by 2 and then a second step that multiplies
it by 1/2. So we have to worry that two different Gauss’s Method reductions
could lead to two different computed values for the determinant.
That is, we must verify that Definition 2.1 gives a well-defined function. The
next two subsections do this, showing that there exists a well-defined function
satisfying the definition.
But first we show that if there is such a function then there is no more than
one. The example above illustrates the idea: we got 5 by following the properties
of the definition. So while we have not yet proved that det 4×4 exists, that there
is a function with properties (1) – (4), if such a function satisfying them does
exist then we know what value it gives on the above matrix.
2.7 Lemma For each n, if there is an n×n determinant function then it is unique.
Proof Suppose that there are two functions det 1 , det 2 : M n×n → R satisfying
the properties of Definition 2.1 and its consequence Lemma 2.4. Given a square
matrix M, fix some way of performing Gauss’s Method to bring the matrix
to echelon form (it does not matter that there are multiple ways, just fix one
of them). By using this fixed reduction as in the above examples — keeping
track of row-scaling factors and how the sign alternates on row swaps, and then
multiplying down the diagonal of the echelon form result — we can compute the
value that these two functions must return on M, and they must return the
same value. Since they give the same output on every input, they are the same
function.
QED
The ‘if there is an n×n determinant function’ emphasizes that, although we
can use Gauss’s Method to compute the only value that a determinant function