Linear Algebra, 2020a

332 Chapter Four. Determinants
each n, the function that such a formula calculates. We will define this function
by a list of properties. We will then prove that a function with these properties
exists and is unique, and also describe how to compute it. (Because we will
eventually prove this, from the start we will just say ‘det(T)’ instead of ‘if there
is a unique determinant function then det(T)’.)
2.1 Definition A n×n determinant is a function det: M n×n → R such that
(1) det(⃗ρ 1 ,...,k· ⃗ρ i + ⃗ρ j ,...,⃗ρ n )=det(⃗ρ 1 ,...,⃗ρ j ,...,⃗ρ n ) for i ≠ j
(2) det(⃗ρ 1 ,...,⃗ρ j ,...,⃗ρ i ,...,⃗ρ n )=−det(⃗ρ 1 ,...,⃗ρ i ,...,⃗ρ j ,...,⃗ρ n ) for i ≠ j
(3) det(⃗ρ 1 ,...,k⃗ρ i ,...,⃗ρ n )=k · det(⃗ρ 1 ,...,⃗ρ i ,...,⃗ρ n ) for any scalar k
(4) det(I) =1 where I is an identity matrix
(the ⃗ρ ’s are the rows of the matrix). We often write |T| for det(T).
2.2 Remark Condition (2) is redundant since
T
ρ i +ρ j
−→
−ρ j +ρ i
−→
ρ i +ρ j
−→
−ρ i
−→
ˆT
swaps rows i and j. We have listed it for consistency with the Gauss’s Method
presentation in earlier chapters.
2.3 Remark Condition (3) does not have a k ≠ 0 restriction, although the Gauss’s
Method operation of multiplying a row by k does have it. The next result shows
that we do not need that restriction here.
2.4 Lemma A matrix with two identical rows has a determinant of zero. A matrix
with a zero row has a determinant of zero. A matrix is nonsingular if and only
if its determinant is nonzero. The determinant of an echelon form matrix is the
product down its diagonal.
Proof To verify the first sentence swap the two equal rows. The sign of the
determinant changes but the matrix is the same and so its determinant is the
same. Thus the determinant is zero.
For the second sentence multiply the zero row by two. That doubles the determinant
but it also leaves the row unchanged, and hence leaves the determinant
unchanged. Thus the determinant must be zero.
Do Gauss-Jordan reduction for the third sentence, T →···→ ˆT. By the first
three properties the determinant of T is zero if and only if the determinant of ˆT is
zero (although the two could differ in sign or magnitude). A nonsingular matrix
T Gauss-Jordan reduces to an identity matrix and so has a nonzero determinant.