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Exercise 8.5.5. Use the following steps to prove that det(AT) = det(A): (1) First write PA = LU as I did in definition 8.3.1. Then take the transpose of both sides. (2) Now take the determinant of your equation with four terms. Analyze which determinants you know, which are simple, and which remain to be thought about. (3) At the end lots of things should cancel and you should get det(A) = ±det (U) = det(AT).

Exercise 8.5.5. Use the following steps to prove that det(AT) = det(A): (1) First write PA = LU as I did in definition 8.3.1. Then take the transpose of both sides. (2) Now take the determinant of your equation with four terms. Analyze which determinants you know, which are simple, and which remain to be thought about. (3) At the end lots of things should cancel and you should get det(A) = ±det (U) = det(AT).

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter11: Matrices And Determinants
Section11.CR: Chapter Review
Problem 11CC
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Exercise 8.5.5. Use the following steps to prove that det(AT) = det(A): (1) First write PA = LU as I did in definition 8.3.1. Then take the transpose of both sides. (2) Now take the determinant of your equation with four terms. Analyze which determinants you know, which are simple, and which remain to be thought about. (3) At the end lots of things should cancel and you should get det(A) = ±det(U) = det(AT).
Transcribed Image Text:Exercise 8.5.5. Use the following steps to prove that det(AT) = det(A): (1) First write PA = LU as I did in definition 8.3.1. Then take the transpose of both sides. (2) Now take the determinant of your equation with four terms. Analyze which determinants you know, which are simple, and which remain to be thought about. (3) At the end lots of things should cancel and you should get det(A) = ±det(U) = det(AT).
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