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Let A be and B be n x n matrices, assume that is an eigenvector of the matrix product AB. Assume that Bx ‡ 0. Prove that Bx is an eigenvector of BA.

Let A be and B be n x n matrices, assume that is an eigenvector of the matrix product AB. Assume that Bx ‡ 0. Prove that Bx is an eigenvector of BA.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.3: Properties Of Determinants
Problem 63E: Let A be an nn matrix in which the entries of each row sum to zero. Find |A|.
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### Problem Statement Let \( A \) and \( B \) be \( n \times n \) matrices, assume that \( \vec{x} \) is an eigenvector of the matrix product \( AB \). Assume that \( B\vec{x} \neq 0 \). Prove that \( B\vec{x} \) is an eigenvector of \( BA \).
Transcribed Image Text:### Problem Statement Let \( A \) and \( B \) be \( n \times n \) matrices, assume that \( \vec{x} \) is an eigenvector of the matrix product \( AB \). Assume that \( B\vec{x} \neq 0 \). Prove that \( B\vec{x} \) is an eigenvector of \( BA \).
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