State True or False and provide a proof or counter example.(a) If n × n matrices A and B have the same reduced row echelon form, then they aresimilar.(b) If A is an n × n matrix and I + A + A² = 0 then A is invertible.(c) If A is an n × n matrix and there exists an n × n matrix B such that AB = I, thenBA= I.(d) If A is an n × n real matrix such that ATA = I then the possible eigenvalues of Aare λ = 1 and λ = -1

As per the guideline, I am supposed to solve only first three subparts of the given question.

ate True or False and provide a proof or counter example. ) If n x n matrices A and B have the same reduced row echelon form, then they are similar. ) If A is an n × n matrix and I + A + A² = 0 then A is invertible. ) If A is an n × n matrix and there exists an n × n matrix B such that AB = I, then BA= I. ) If A is an n × n real matrix such that ATA = I then the possible eigenvalues of A are λ = 1 and λ = -1

ate True or False and provide a proof or counter example. ) If n x n matrices A and B have the same reduced row echelon form, then they are similar. ) If A is an n × n matrix and I + A + A² = 0 then A is invertible. ) If A is an n × n matrix and there exists an n × n matrix B such that AB = I, then BA= I. ) If A is an n × n real matrix such that ATA = I then the possible eigenvalues of A are λ = 1 and λ = -1

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

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State True or False and provide a proof or counter example.
(a) If n × n matrices A and B have the same reduced row echelon form, then they are
similar.
(b) If A is an n × n matrix and I + A + A² = 0 then A is invertible.
(c) If A is an n × n matrix and there exists an n × n matrix B such that AB = I, then
BA= I.
(d) If A is an n × n real matrix such that ATA = I then the possible eigenvalues of A
are λ = 1 and λ = -1

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