True or False? Determine if the following statements are true or false. If true, give a proof. If false, explain why and/or provide a counter-example. (When providing a counter-example, be sure to explain why your example is in fact a counter-example.) 1) "Let A E Mnxn(F). If A has no eigenvalues, then CA(T) is irreducible." 2) "Let A E Mnxn(F). If LA : F" →F is not surjective, then A = 0 is an eigenvalue for A." 3) 'Let A, B E Mnxn (F). If ABis invertible, then A and B must both be invertible."

True or False? Determine if the following statements are true or false. If true, give a proof. If false, explain why and/or provide a counter-example. (When providing a counter-example, be sure to explain why your example is in fact a counter-example.) 1) "Let A E Mnxn(F). If A has no eigenvalues, then CA(T) is irreducible." 2) "Let A E Mnxn(F). If LA : F" →F is not surjective, then A = 0 is an eigenvalue for A." 3) 'Let A, B E Mnxn (F). If ABis invertible, then A and B must both be invertible."

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 59EQ

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