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 graded for 20 points. (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-A¹ is skew-symmetric. (9) If A and B are symmetric, then so is AB. (10) If A = (aj)nxn is skew-symmetric, then a = 0 for all 1 ≤ i ≤ n. (11) tr(A + B) = tr(A) + tr(B) (12) tr(aA) = atr(A), where a € 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.
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 graded for 20 points. (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-A¹ is skew-symmetric. (9) If A and B are symmetric, then so is AB. (10) If A = (aj)nxn is skew-symmetric, then a = 0 for all 1 ≤ i ≤ n. (11) tr(A + B) = tr(A) + tr(B) (12) tr(aA) = atr(A), where a € 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.
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
Number 16
Transcribed Image Text: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 graded for 20 points.
(1) If AB
(2) If A²
(3) AB =
(4) If AB AC, then B = C.
O, then A= O or B=0.
I, then A= I or A = -1.
BA
=
(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 = (aj)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 R and A has size n x 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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