Linear Algebra

Section I. Definition 319Exercise 14 shows that sgn(φ −1 ) = sgn(φ). Since every permutation is theinverse of another, a sum over all inverses φ −1 is a sum over all permutations= ∑s 1,σ ( 1) . . . s n,σ(n) sgn(σ) = ∣ ∣ Ttrans perms σas required.QEDExercisesThese summarize the notation used in this book for the 2- and 3- permutations.i 1 2 i 1 2 3φ 1 (i) 1 2 φ 1 (i) 1 2 3φ 2 (i) 2 1 φ 2 (i) 1 3 2φ 3 (i) 2 1 3φ 4 (i) 2 3 1φ 5 (i) 3 1 2φ 6 (i) 3 2 14.9 Give the permutation expansion of a general 2×2 matrix and its transpose.̌ 4.10 This problem appears also in the prior subsection.(a) Find the inverse of each 2-permutation.(b) Find the inverse of each 3-permutation.̌ 4.11 (a) Find the signum of each 2-permutation.(b) Find the signum of each 3-permutation.4.12 Find the only nonzero term in the permutation expansion of this matrix.0 1 0 01 0 1 00 1 0 1∣0 0 1 0∣Compute that determinant by finding the signum of the associated permutation.4.13 [Strang 80] What is the signum of the n-permutation φ = 〈n, n − 1, . . . , 2, 1〉?4.14 Prove these.(a) Every permutation has an inverse.(b) sgn(φ −1 ) = sgn(φ)(c) Every permutation is the inverse of another.4.15 Prove that the matrix of the permutation inverse is the transpose of the matrixof the permutation P φ −1 = P φ trans , for any permutation φ.̌ 4.16 Show that a permutation matrix with m inversions can be row swapped to theidentity in m steps. Contrast this with Corollary 4.4.̌ 4.17 For any permutation φ let g(φ) be the integer defined in this way.g(φ) = ∏ i