16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.3.12

Determinants

Mathematics — Linear Algebra · permutation-matrices

Problem

\sigma

is a permutation of degree

n

and

A

is an

n \times n

matrix over the field

F

with row vectors

\alpha_1, \ldots, \alpha_n

, let

\sigma(A)

denote the

n \times n

matrix with row vectors

\alpha_{\sigma 1}, \ldots, \alpha_{\sigma n}

. ((a)) Prove that

\sigma(AB) = \sigma(A)B

, and in particular that

\sigma(A) = \sigma(I)A

. ((b)) If

T

is the linear operator of Exercise 9, prove that the matrix of

T

in the standard ordered basis is

\sigma(I)

. ((c)) Is

\sigma^{-1}(I)

the inverse matrix of

\sigma(I)

? ((d)) Is it true that

\sigma(A)

is similar to

A