Linear Algebra, 2020a

350 Chapter Four. Determinants
Proof We must check that it satisfies the four conditions from the definition of
determinant, Definition 2.1.
Condition (4) is easy: where I is the n×n identity, in
d(I) =
∑
ι 1,φ(1) ι 2,φ(2) ···ι n,φ(n) sgn(φ)
perm φ
all of the terms in the summation are zero except for the one where the permutation
φ is the identity, which gives the product down the diagonal, which is
one.
For condition (3) suppose that T
kρ i
−→ ˆT and consider d(ˆT).
∑
ˆt 1,φ(1) ···ˆt i,φ(i) ···ˆt n,φ(n) sgn(φ)
perm φ
= ∑ t 1,φ(1) ···kt i,φ(i) ···t n,φ(n) sgn(φ)
φ
Factor out k to get the desired equality.
= k · ∑
t 1,φ(1) ···t i,φ(i) ···t n,φ(n) sgn(φ) =k · d(T)
φ
ρ i↔ρ j
−→
For (2) suppose that T ˆT. We must show that d(ˆT) is the negative
of d(T).
d(ˆT) =
∑ ˆt 1,φ(1) ···ˆt i,φ(i) ···ˆt j,φ(j) ···ˆt n,φ(n) sgn(φ) (∗)
perm φ
We will show that each term in (∗) is associated with a term in d(T), and that the
two terms are negatives of each other. Consider the matrix from the multilinear
expansion of d(ˆT) giving the term ˆt 1,φ(1) ···ˆt i,φ(i) ···ˆt j,φ(j) ···ˆt n,φ(n) sgn(φ).
⎛
⎜
⎝
.
ˆt i,φ(i)
. ..
ˆt j,φ(j)
It is the result of the ρ i ↔ ρ j operation performed on this matrix.
⎛
⎞
.
t i,φ(j)
.
⎜ t
⎝ j,φ(i)
⎟
⎠
. ..
.
⎞
⎟
⎠