Linear Algebra

318 Chapter 4. Determinants
4.10 Give the permutation expansion of a general 2×2 matrix and its transpose.
� 4.11 This problem appears also in the prior subsection.
(a) Find the inverse of each 2-permutation.
(b) Find the inverse of each 3-permutation.
� 4.12 (a) Find the signum of each 2-permutation.
(b) Find the signum of each 3-permutation.
4.13 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 matrix
of the permutation Pφ−1 = Pφ trans , for any permutation φ.
� 4.16 Show that a permutation matrix with m inversions can be row swapped to
the identity in m steps. Contrast this with Corollary 4.6.
� 4.17 For any permutation φ let g(φ) be the integer defined in this way.
�
g(φ) = [φ(j) − φ(i)]
i