Question 3 Which of the following are true for all invertible matrices A, B, C of the same size? Find a proof or counterexample in each case. -1 (a) (ABC) 1 A-¹B-¹C-1 -1 = (b) (ABA) = A-¹B-¹ A-1 -1 (c) (A¹) A = (d) −(-A)¯¹ = A−¹ (e) C-1 (ABC-1)−¹ AB = I -1 (f) (A+B)¹ = A−¹ + B−¹ (g) A¹(IA) A = A + I 1 (h) (AI) (A-¹-I) = A-¹ - A (i) A22A+ I = 0⇒ A¯¹ = 2I – A (j) A² - 2A+ I = 0⇒ A = I
Question 3 Which of the following are true for all invertible matrices A, B, C of the same size? Find a proof or counterexample in each case. -1 (a) (ABC) 1 A-¹B-¹C-1 -1 = (b) (ABA) = A-¹B-¹ A-1 -1 (c) (A¹) A = (d) −(-A)¯¹ = A−¹ (e) C-1 (ABC-1)−¹ AB = I -1 (f) (A+B)¹ = A−¹ + B−¹ (g) A¹(IA) A = A + I 1 (h) (AI) (A-¹-I) = A-¹ - A (i) A22A+ I = 0⇒ A¯¹ = 2I – A (j) A² - 2A+ I = 0⇒ A = I
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Transcribed Image Text:Question 3
Which of the following are true for all invertible matrices A, B, C of the same size? Find a proof or
counterexample in each case.
-1
(a) (ABC) 1 A-¹B-¹C-1
-1
=
(b) (ABA) = A-¹B-¹ A-1
-1
(c) (A¹) A
=
(d) −(-A)¯¹ = A−¹
(e) C-1 (ABC-1)−¹ AB = I
-1
(f) (A+B)¹ = A−¹ + B−¹
(g) A¹(IA) A = A + I
1
(h) (AI) (A-¹-I) = A-¹ - A
(i) A22A+ I = 0⇒ A¯¹ = 2I – A
(j) A² - 2A+ I = 0⇒ A = I
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