11 to 16 Let A, B and C be matrices such that the following operations are defined. Let A andtr(A) denote the transpose and trace of A respectively. Let I and O denote identity andzero matrices respectively. For each of the following statements, determine whether it isTrue or False. If it is True, prove it. If it is False, give a counter example to disprove it..Four or five randomly selected statements will be(1) If AB = O, then A= O or B = 0.(2) If A² = I, then A = I or A = -1.(3) AB=BA(4) If AB AC, then B = C.(5) A(BC) = (AB)C(6) (AB) = B' A'(7) A+ A' is symmetric.(8) A A' is skew-symmetric.(9) If A and B are symmetric, then so is AB.(10) If A = (ai)nxn is skew-symmetric, then a=0 for all 1 ≤ i ≤ n.(11) tr(A + B) = tr(A) + tr(B)(12) tr(aA) = a"tr(A), where a E R and A has size nx n.(13) tr(AB) = tr(A)tr(B)-(14) tr(AB) = tr(BA)(15) tr(AA) = (tr(A))2(16) There exist square matrices A and B such that AB - BA= I.

According Bartleby guidelines we can solve only 3 subpart rest of the questions can be reposted

Answered: (11) tr(A + B)= tr(A) + tr(B) (12)…

(11) tr(A + B)= tr(A) + tr(B) (12) tr(aA) = a"tr(A), where a E R and A has size nxn. (13) tr(AB) = tr(A)tr(B) (14) tr(AB) = tr(BA)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 31E

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Question

11 to 16

Let A, B and C be matrices such that the following operations are defined. Let A and
tr(A) denote the transpose and trace of A respectively. Let I and O denote identity and
zero matrices respectively. For each of the following statements, determine whether it is
True or False. If it is True, prove it. If it is False, give a counter example to disprove it..
Four or five randomly selected statements will be
(1) If AB = O, then A= O or B = 0.
(2) If A² = I, then A = I or A = -1.
(3) AB=BA
(4) If AB AC, then B = C.
(5) A(BC) = (AB)C
(6) (AB) = B' A'
(7) A+ A' is symmetric.
(8) A A' is skew-symmetric.
(9) If A and B are symmetric, then so is AB.
(10) If A = (ai)nxn is skew-symmetric, then a=0 for all 1 ≤ i ≤ n.
(11) tr(A + B) = tr(A) + tr(B)
(12) tr(aA) = a"tr(A), where a E R and A has size nx n.
(13) tr(AB) = tr(A)tr(B)
-
(14) tr(AB) = tr(BA)
(15) tr(AA) = (tr(A))2
(16) There exist square matrices A and B such that AB - BA= I.

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