Definition:If A, B E Matnxn (C), we say A and B are conjugate if there is an invertible matrix CE Matnxn (C)such that C-¹ AC = B.3.a) Prove that the relation~ defined by A~ B if and only if A and B are conjugate is an equivalencerelation on the set Matnxn (C).b) If A and B are conjugate, prove that PÂ(t) = PÂ(t), that is, A and B have the same characteristicpolynomial.c) If A and B are conjugate, prove det(A) = det (B). Give an example to show that the conversedoes not hold.d) Show that there are at least n equivalences classes of the relation~ that consist of matrices withdeterminant 0.(Hint: You may use part (b) even if you did not solve it.)

As per our guidelines we are supposed to answer only three subpart. Here first 3 sub

3. a) Prove that the relation~ defined by A~ B if and only if A and B are conjugate is an equivalence relation on the set Matnxn (C). b) If A and B are conjugate, prove that PA(t) = PB (t), that is, A and B have the same characteristic polynomial. c) If A and B are conjugate, prove det(A) = det (B). Give an example to show that the converse does not hold. d) Show that there are at least n equivalences classes of the relation ~that consist of matrices with determinant 0.

3. a) Prove that the relation~ defined by A~ B if and only if A and B are conjugate is an equivalence relation on the set Matnxn (C). b) If A and B are conjugate, prove that PA(t) = PB (t), that is, A and B have the same characteristic polynomial. c) If A and B are conjugate, prove det(A) = det (B). Give an example to show that the converse does not hold. d) Show that there are at least n equivalences classes of the relation ~that consist of matrices with determinant 0.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 76EQ

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