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3. Let A € R¹x and aij denote the element in ith row and jth column. If 0 ≤ aij < 1 Vi € 72 {1,...,n}, j = {1,..., n}, and Σ aij < 1 for i = 1, ..., n, then p(A) < 1 where p(A) is the spectral radius of A.

3. Let A € R¹x and aij denote the element in ith row and jth column. If 0 ≤ aij < 1 Vi € 72 {1,...,n}, j = {1,..., n}, and Σ aij < 1 for i = 1, ..., n, then p(A) < 1 where p(A) is the spectral radius of A.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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Please solve part 3 of q1 but please correctly and according to the question requirement

1 Problem 1 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 |A₂| 1 where |A₁| denotes the absolute value of A₂ for i = 1,..., n. 3. Let A € Rxn and aij denote the element in th row and th column. If 0 ≤ aij < 1 Vi € TL {1,...,n}, j = {1,..., n}, and Σ aij < 1 for i = 1,..., n, then p(A) < 1 where p(A) is the spectral radius of A. 2 Problem 2 Let A = aßT where a, B E R and B¹ a 0. 1. When will the linear system = Ar be asymptotically or marginally stable? 2. Compute et
Transcribed Image Text:1 Problem 1 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 |A₂| 1 where |A₁| denotes the absolute value of A₂ for i = 1,..., n. 3. Let A € Rxn and aij denote the element in th row and th column. If 0 ≤ aij < 1 Vi € TL {1,...,n}, j = {1,..., n}, and Σ aij < 1 for i = 1,..., n, then p(A) < 1 where p(A) is the spectral radius of A. 2 Problem 2 Let A = aßT where a, B E R and B¹ a 0. 1. When will the linear system = Ar be asymptotically or marginally stable? 2. Compute et
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