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). Definition 8.3.1 (Reduction Algorithm). Given a square matrix A, we can apply the reduction algorithm to obtain P.A = L U where U is an upper triangular matrix, L is a lower triangular matrix with 1's on the diagonal, and P is a permutation matrix. Then det(P). det(A) = ±det(A): det (L) det (U)= det(U) = u₁1... Unn. =
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). Definition 8.3.1 (Reduction Algorithm). Given a square matrix A, we can apply the reduction algorithm to obtain P.A = L U where U is an upper triangular matrix, L is a lower triangular matrix with 1's on the diagonal, and P is a permutation matrix. Then det(P). det(A) = ±det(A): det (L) det (U)= det(U) = u₁1... Unn. =
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 7AEXP
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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).
Definition 8.3.1 (Reduction Algorithm). Given a square
matrix A, we can apply the reduction algorithm to obtain
P.A = LU where U is an upper triangular matrix, Lis
a lower triangular matrix with 1's on the diagonal, and P is
+det(A)
a permutation matrix. Then det(P) · det(A)
det (L) det (U) = det(U) = U₁1 ... Unn.
=
=
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