Linear Algebra, 2020a

Section I. Definition 353
in the top and t 3,1 t 1,2 t 2,3 is in the bottom. That’s perfectly sensible — the six
in the top arise from all of the ways of picking one entry of T from each row and
column while the six in the bottom are all of the ways of picking one entry of T
from each column and row, so of course they are the same set.
Next observe that in the two expansions, each t-product expression is not
necessarily associated with the same permutation matrix. For instance, on the
top t 1,2 t 2,3 t 3,1 is associated with the matrix for the map 1 ↦→ 2, 2 ↦→ 3, 3 ↦→ 1.
On the bottom t 3,1 t 1,2 t 2,3 is associated with the matrix for the map 1 ↦→ 3,
2 ↦→ 1, 3 ↦→ 2. The second map is inverse to the first. This is also perfectly
sensible — both the matrix transpose and the map inverse flip the 1, 2 to 2, 1,
flip the 2, 3 to 3, 2, and flip 3, 1 to 1, 3.
We finish by noting that the determinant of P φ equals the determinant of
P φ
−1, as Exercise 16 shows.
QED
Exercises
These summarize the notation used in this book for the 2- and 3-permutations.
i 1 2
φ 1 (i) 1 2
φ 2 (i) 2 1
i 1 2 3
φ 1 (i) 1 2 3
φ 2 (i) 1 3 2
φ 3 (i) 2 1 3
φ 4 (i) 2 3 1
φ 5 (i) 3 1 2
φ 6 (i) 3 2 1
4.11 Give the permutation expansion of a general 2×2 matrix and its transpose.
̌ 4.12 This problem appears also in the prior subsection.
(a) Find the inverse of each 2-permutation.
(b) Find the inverse of each 3-permutation.
̌ 4.13 (a) Find the signum of each 2-permutation.
(b) Find the signum of each 3-permutation.
4.14 Find the only nonzero term in the permutation expansion of this matrix.
0 1 0 0
1 0 1 0
0 1 0 1
∣
0 0 1 0
∣
Compute that determinant by finding the signum of the associated permutation.
4.15 [Strang 80] What is the signum of the n-permutation φ = 〈n, n − 1,...,2,1〉?
4.16 Prove these.
(a) Every permutation has an inverse.
(b) sgn(φ −1 )=sgn(φ)
(c) Every permutation is the inverse of another.
4.17 Prove that the matrix of the permutation inverse is the transpose of the matrix
of the permutation P φ −1 = P T φ , for any permutation φ.