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6. For the symmetric matrix A = 122 200 202 , you are given the following information about A: • det(A-XI) = (A − 1)(A+2)(A-4) ● (2,1,2), (1,2,-2), and (-2, 2, 1) are eigenvectors. a.) Find the eigenvalues of each of the vectors (2, 1, 2), (1, 2,-2), and (-2,2,1). b.) Find an orthonormal basis of R³ consisting of eigenvectors of the matrix A. c.) State an orthogonal matrix Q and a diagonal matrix D for which QT AQ = D.

6. For the symmetric matrix A = 122 200 202 , you are given the following information about A: • det(A-XI) = (A − 1)(A+2)(A-4) ● (2,1,2), (1,2,-2), and (-2, 2, 1) are eigenvectors. a.) Find the eigenvalues of each of the vectors (2, 1, 2), (1, 2,-2), and (-2,2,1). b.) Find an orthonormal basis of R³ consisting of eigenvectors of the matrix A. c.) State an orthogonal matrix Q and a diagonal matrix D for which QT AQ = D.

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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6. For the symmetric matrix A = 12 2 200 202 you are given the following information about A: • det(A-XI) = (A-1)(A + 2)(AX - 4) (2, 1, 2), (1, 2,-2), and (-2, 2, 1) are eigenvectors. a.) Find the eigenvalues of each of the vectors (2, 1, 2), (1, 2, -2), and (-2, 2, 1). b.) Find an orthonormal basis of R³ consisting of eigenvectors of the matrix A. c.) State an orthogonal matrix Q and a diagonal matrix D for which QTAQ = D.
Transcribed Image Text:6. For the symmetric matrix A = 12 2 200 202 you are given the following information about A: • det(A-XI) = (A-1)(A + 2)(AX - 4) (2, 1, 2), (1, 2,-2), and (-2, 2, 1) are eigenvectors. a.) Find the eigenvalues of each of the vectors (2, 1, 2), (1, 2, -2), and (-2, 2, 1). b.) Find an orthonormal basis of R³ consisting of eigenvectors of the matrix A. c.) State an orthogonal matrix Q and a diagonal matrix D for which QTAQ = D.
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