Kunze Linear Algebra · Problem 5.3.12

Determinants

Mathematics — Methods · permutations-matrices

Problem

If σ\sigma is a permutation of degree nn and AA is an n×nn \times n matrix over the field FF with row vectors α1,…,αn\alpha_1, \ldots, \alpha_n, let σ(A)\sigma(A) denote the n×nn \times n matrix with row vectors ασ1,…,ασn\alpha_{\sigma 1}, \ldots, \alpha_{\sigma n}. ((a)) Prove that σ(AB)=σ(A)B\sigma(AB) = \sigma(A)B, and in particular that σ(A)=σ(I)A\sigma(A) = \sigma(I)A. ((b)) If TT is the linear operator of Exercise 9, prove that the matrix of TT in the standard ordered basis is σ(I)\sigma(I). ((c)) Is σ−1(I)\sigma^{-1}(I) the inverse matrix of σ(I)\sigma(I)? ((d)) Is it true that σ(A)\sigma(A) is similar to AA?
Intuitive explanation

The key insight is that permuting rows is a linear operation that can be represented as left-multiplication by a permutation matrix. When you rearrange rows of AA, you're essentially applying a fixed transformation that doesn't depend on AA itself—only on which rows go where. This means row permutation distributes nicely over matrix multiplication, and the permutation matrix σ(I)\sigma(I) captures the entire permutation as a linear map. Understanding how permutation matrices behave (especially their invertibility and relationship to similarity) will unlock all four parts.

Intuitive explanation

The key insight is that permuting rows is a linear operation that can be represented as left-multiplication by a permutation matrix. When you rearrange rows of AA, you're essentially applying a fixed transformation that doesn't depend on AA itself—only on which rows go where. This means row permutation distributes nicely over matrix multiplication, and the permutation matrix σ(I)\sigma(I) captures the entire permutation as a linear map. Understanding how permutation matrices behave (especially their invertibility and relationship to similarity) will unlock all four parts.

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