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Are the following statements true or false? If true, prove the statement. If false, give a counterexample. 1. Let A € R³×3. If each eigenvalue of A is zero, then A² = 0. (Note: 0 denotes the zero matrix € R³x3)

Are the following statements true or false? If true, prove the statement. If false, give a counterexample. 1. Let A € R³×3. If each eigenvalue of A is zero, then A² = 0. (Note: 0 denotes the zero matrix € R³x3)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Are the following statements true or false? If true, prove the statement. If false, give a counterexample. 1. Let A € R³×³. If each eigenvalue of A is zero, then A² = 0. (Note: 0 denotes the zero matrix € R³x3) 2. Let A € Rnxn be a real symmetric matrix, and A₁, A2,..., An the eigenvalues of A, then || A||2 = max |A₁| where A denotes the absolute value of A, for i = 1,..., n.
Transcribed Image Text:Are the following statements true or false? If true, prove the statement. If false, give a counterexample. 1. Let A € R³×³. If each eigenvalue of A is zero, then A² = 0. (Note: 0 denotes the zero matrix € R³x3) 2. Let A € Rnxn be a real symmetric matrix, and A₁, A2,..., An the eigenvalues of A, then || A||2 = max |A₁| where A denotes the absolute value of A, for i = 1,..., n.
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