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(a) Suppose that A and B are square matrices of the same size. Prove that the (A + B)² = A² + 2AB + B² is satisfied if and only if A and B commute. (Note that you are being asked for a general proof. It is not enough to just choose two specific matrices.) Exercise 3. condition

(a) Suppose that A and B are square matrices of the same size. Prove that the (A + B)² = A² + 2AB + B² is satisfied if and only if A and B commute. (Note that you are being asked for a general proof. It is not enough to just choose two specific matrices.) Exercise 3. condition

Algebra and Trigonometry (6th Edition)
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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(a) Suppose that A and B are square matrices of the same size. Prove that the (A + B)² = A² + 2AB + B² is satisfied if and only if A and B commute. (Note that you are being asked for a general proof. It is not enough to just choose two specific matrices.) (b) Suppose that A is a square matrix with the property that A³ = 0 Exercise 3. condition (here "O" means the zero matrix of the same size as A). Prove that the matrix I + A is invertible, and that its inverse is given by (I + A)−¹ = I − A + A². (Note that you cannot assume anything about A other than the condition A³ = 0.) (c) Suppose that A is an invertible matrix. Prove that AT is also invertible, and that its inverse is given by (AT)−¹ = (A¯¹)T . (Hint: you need to verify the condition in the definition of the "inverse" of a matrix. The property (AB)T = BT AT might be useful.)
Transcribed Image Text:(a) Suppose that A and B are square matrices of the same size. Prove that the (A + B)² = A² + 2AB + B² is satisfied if and only if A and B commute. (Note that you are being asked for a general proof. It is not enough to just choose two specific matrices.) (b) Suppose that A is a square matrix with the property that A³ = 0 Exercise 3. condition (here "O" means the zero matrix of the same size as A). Prove that the matrix I + A is invertible, and that its inverse is given by (I + A)−¹ = I − A + A². (Note that you cannot assume anything about A other than the condition A³ = 0.) (c) Suppose that A is an invertible matrix. Prove that AT is also invertible, and that its inverse is given by (AT)−¹ = (A¯¹)T . (Hint: you need to verify the condition in the definition of the "inverse" of a matrix. The property (AB)T = BT AT might be useful.)
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