13. Give examples of matrices satisfying the properties below. (a) A has size 2 × 2 and 1,2 are eigenvalues of A. (b) A has size 3 × 3, the only eigenvalues of A are 4 and –6, and nullity(A+ 6I3) = 2. (c) A has size 3 x 3, the only eigenvalue of A is –6, and nullity(A+6I3) = 2.
13. Give examples of matrices satisfying the properties below. (a) A has size 2 × 2 and 1,2 are eigenvalues of A. (b) A has size 3 × 3, the only eigenvalues of A are 4 and –6, and nullity(A+ 6I3) = 2. (c) A has size 3 x 3, the only eigenvalue of A is –6, and nullity(A+6I3) = 2.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 41EQ
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![13. Give examples of matrices satisfying the properties below.
(a) A has size 2 × 2 and 1,2 are eigenvalues of A.
(b) A has size 3 x 3, the only eigenvalues of A are 4 and -6, and nullity(A+ 6I3) = 2.
(c) A has size 3 x 3, the only eigenvalue of A is –6, and nullity(A+6I3) = 2.
(d) The characteristic polynomial of A is –t(t² – t+6).
(e) The characteristic polynomial of A is (1 – t)(t² +2).
(f) The characteristic polynomial of A is (4 – t)(2 – t)³ and rank(A – 214) = 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdbd9345f-cdaf-40a0-9604-7a9d3647c236%2F31ecb4c5-68a1-4730-84cd-f478987f9253%2F6stp26b_processed.png&w=3840&q=75)
Transcribed Image Text:13. Give examples of matrices satisfying the properties below.
(a) A has size 2 × 2 and 1,2 are eigenvalues of A.
(b) A has size 3 x 3, the only eigenvalues of A are 4 and -6, and nullity(A+ 6I3) = 2.
(c) A has size 3 x 3, the only eigenvalue of A is –6, and nullity(A+6I3) = 2.
(d) The characteristic polynomial of A is –t(t² – t+6).
(e) The characteristic polynomial of A is (1 – t)(t² +2).
(f) The characteristic polynomial of A is (4 – t)(2 – t)³ and rank(A – 214) = 3.
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