(a) Suppose that A and B are square matrices of the same size. Prove that the(A + B)² = A² + 2AB + B²is satisfied if and only if A and B commute. (Note that you are being asked for a generalproof. It is not enough to just choose two specific matrices.)(b) Suppose that A is a square matrix with the property thatA³ = 0Exercise 3.condition(here "O" means the zero matrix of the same size as A). Prove that the matrix I + A isinvertible, 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 inverseis given by(AT)−¹ = (A¯¹)T .(Hint: you need to verify the condition in the definition of the "inverse" of a matrix. Theproperty (AB)T = BT AT might be useful.)

We have to show the given condition is satisfied if and only if A and B commute.According to our…

(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)

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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Question

(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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