Linear Algebra, 2020a

Section I. Definition 349
Thus, in aggregate, the number of inversions changes from even to odd, or from
oddtoeven.
QED
4.5 Corollary If a permutation matrix has an odd number of inversions then
swapping it to the identity takes an odd number of swaps. If it has an even
number of inversions then swapping to the identity takes an even number.
Proof The identity matrix has zero inversions. To change an odd number to
zero requires an odd number of swaps, and to change an even number to zero
requires an even number of swaps.
QED
4.6 Example The matrix in Example 4.3 can be brought to the identity with one
swap ρ 1 ↔ ρ 3 . (So the number of swaps needn’t be the same as the number of
inversions, but the oddness or evenness of the two numbers is the same.)
4.7 Definition The signum of a permutation sgn(φ) is −1 if the number of
inversions in φ is odd and is +1 if the number of inversions is even.
4.8 Example Using the notation for the 3-permutations from Example 3.8 we
have
⎛
⎜
1 0 0
⎞
⎛
⎟
⎜
1 0 0
⎞
⎟
P φ1 = ⎝0 1 0⎠ P φ2 = ⎝0 0 1⎠
0 0 1
0 1 0
so sgn(φ 1 )=1 because there are no inversions, while sgn(φ 2 )=−1 because
there is one.
We still have not shown that the determinant function is well-defined because
we have not considered row operations on permutation matrices other than row
swaps. We will finesse this issue. Define a function d: M n×n → R by altering
the permutation expansion formula, replacing |P φ | with sgn(φ).
d(T) =
∑
permutations φ
t 1,φ(1) t 2,φ(2) ···t n,φ(n) · sgn(φ)
The advantage of this formula is that the number of inversions is clearly welldefined
— just count them. Therefore, we will be finished showing that an n×n
determinant function exists when we show that d satisfies the conditions in the
definition of a determinant.
4.9 Lemma The function d above is a determinant. Hence determinant functions
det n×n exist for every n.