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(a) Use matrix multiplication to show that x1 = and x2 are -2 -1 eigenvectors of A. What are the corresponding eigenvalues?

(a) Use matrix multiplication to show that x1 = and x2 are -2 -1 eigenvectors of A. What are the corresponding eigenvalues?

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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3 -2 1 Consider the matrix A = 3 0 -2 1 (a) Use matrix multiplication to show that x = and x2 = are -2 eigenvectors of A. What are the corresponding eigenvalues? (b) Using your answer to (a) and either the trace or the determinant of A, find the third eigenvalue, and a corresponding eigenvector x3. Scale x3 so that its entries are integers. (c) Verify that {x1, x2, x3} is a linearly independent set. (d) Is A diagonalisable? If so, write down a matrix P which will diagonalise A. (e) Consider the system of first ODEŚ: dx 3.x 2y + z 3y -2x + 2y dt (i) Use parts (a) and (b) or solve otherwise to find the unknown constants r, y and z I| || ||
Transcribed Image Text:3 -2 1 Consider the matrix A = 3 0 -2 1 (a) Use matrix multiplication to show that x = and x2 = are -2 eigenvectors of A. What are the corresponding eigenvalues? (b) Using your answer to (a) and either the trace or the determinant of A, find the third eigenvalue, and a corresponding eigenvector x3. Scale x3 so that its entries are integers. (c) Verify that {x1, x2, x3} is a linearly independent set. (d) Is A diagonalisable? If so, write down a matrix P which will diagonalise A. (e) Consider the system of first ODEŚ: dx 3.x 2y + z 3y -2x + 2y dt (i) Use parts (a) and (b) or solve otherwise to find the unknown constants r, y and z I| || ||
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