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5. For each of the statements given below decide if it is true or false. If it is true explain why. If it is false give a counterexample. a) If u, v are vectors in R2 which satisfy ||u + v|| = ||uv|| then u must be orthogonal to v. b) If A is a 3 x 3 matrix which is both symmetric and orthogonal then A³ = A. c) If A is an n x n matrix and v is eigenvector of A, then v is also an eigenvector of A². d) If A is an n x n matrix and λ is eigenvalue of A, then λ is also an eigenvalue of A².

5. For each of the statements given below decide if it is true or false. If it is true explain why. If it is false give a counterexample. a) If u, v are vectors in R2 which satisfy ||u + v|| = ||uv|| then u must be orthogonal to v. b) If A is a 3 x 3 matrix which is both symmetric and orthogonal then A³ = A. c) If A is an n x n matrix and v is eigenvector of A, then v is also an eigenvector of A². d) If A is an n x n matrix and λ is eigenvalue of A, then λ is also an eigenvalue of A².

Advanced Engineering Mathematics
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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5. For each of the statements given below decide if it is true or false. If it is true explain why. If it is false give a counterexample. a) If u, v are vectors in R2 which satisfy ||u + v|| = ||uv|| then u must be orthogonal to v. b) If A is a 3 x 3 matrix which is both symmetric and orthogonal then A³ = A. c) If A is an n x n matrix and v is eigenvector of A, then v is also an eigenvector of A². d) If A is an n x n matrix and is eigenvalue of A, then λ is also an eigenvalue of A².
Transcribed Image Text:5. For each of the statements given below decide if it is true or false. If it is true explain why. If it is false give a counterexample. a) If u, v are vectors in R2 which satisfy ||u + v|| = ||uv|| then u must be orthogonal to v. b) If A is a 3 x 3 matrix which is both symmetric and orthogonal then A³ = A. c) If A is an n x n matrix and v is eigenvector of A, then v is also an eigenvector of A². d) If A is an n x n matrix and is eigenvalue of A, then λ is also an eigenvalue of A².
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