Prove that if A2 = O, then 0 is the only eigenvalue of A.Getting Started: You need to show that if there exists a nonzero vector x and a real number λ such that Ax = λx, then if A2 = O, λ must be zero.(i) A2 = A ∙ A, so you can write A2x as A(Ax).(ii) Use the fact that Ax = λx and the properties of matrix multiplication to show that A2x = λ2x.(iii) A2 is a zero matrix, so you can conclude that λ must be zero.

To show that if A2=O then 0 is the only eigenvalue of A. We need to show that if there exists a non…

Answered: Prove that if A2 = O, then 0 is the…

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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Prove that if A² = O, then 0 is the only eigenvalue of A.
Getting Started: You need to show that if there exists a nonzero vector x and a real number λ such that Ax = λx, then if A² = O, λ must be zero.
(i) A² = A ∙ A, so you can write A²x as A(Ax).
(ii) Use the fact that Ax = λx and the properties of matrix multiplication to show that A²x = λ²x.
(iii) A² is a zero matrix, so you can conclude that λ must be zero.

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