Let A and B be n x n matrices. Let P be an invertible matrix such that A = PBP-1. Let TA : R" → R", TA(T) = Au and Tg : R" → R", T;(7) = Bữ (a) Prove that det(A) = det B. (b) Let A E R". Prove that A is an eigenvalue of TA if and only if A is an eigenvalue of TB. (c) Prove that if a E ker(TB), then Pã e ker(TA).

Let A and B be n x n matrices. Let P be an invertible matrix such that A = PBP-1. Let TA : R" → R", TA(T) = Au and Tg : R" → R", T;(7) = Bữ (a) Prove that det(A) = det B. (b) Let A E R". Prove that A is an eigenvalue of TA if and only if A is an eigenvalue of TB. (c) Prove that if a E ker(TB), then Pã e ker(TA).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let A and B be n x n matrices. Let P be an invertible matrix such that A = PBP-1.
Let TA : R" → R", TA(T) = Au and Tg : R" → R", T;(7) = Bữ
(a) Prove that det(A) = det B.
(b) Let A E R". Prove that A is an eigenvalue of TA if and only if A is an
eigenvalue of TB.
(c) Prove that if T e ker(TB), then PI e ker(TA).

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