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2. Use mathematical induction to show that if a is an eigenvalue of an n × n matrix A, with x a corresponding eigenvector, then, for each positive integer m, 2m is an eigenvalue of A™, with x a corresponding eigenvector

2. Use mathematical induction to show that if a is an eigenvalue of an n × n matrix A, with x a corresponding eigenvector, then, for each positive integer m, 2m is an eigenvalue of A™, with x a corresponding eigenvector

Linear Algebra: A Modern Introduction
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 24EQ
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solve question 2 with explanation asap within 15 min or else downvote
1. Suppose x is an eigenvector of A corresponding to an eigenvalue 1. Show that x is an eigenvector of 51 – A. What is the corresponding eigenvalue 2? 2. Use mathematical induction to show that if 2 is an eigenvalue of an n x n matrix A, with x a corresponding eigenvector, then, for each positive integer m, 2m is an eigenvalue of A", with x a corresponding eigenvector 3. Show that if x is an eigenvector of the matrix product AB and Bx # 0, then Bx is an eigenvector of BA. ГО.4 —0.3 Lo.4 -0.5 -0.75] 4. Let A = . Explain why Ak approaches as k → o. 1.2 1.0 1.50 5. Suppose A = PDP-1, where P is 2 x 2 and D = 6 91: Let B = 51 – 3A + A². Show that B is diagonalizable by finding a suitable factorization of B. 6. If a, b, andc are distinct numbers, then the following system is inconsistent because the graphs of the equations are parallel planes. Show that the set of all least-squares solutions of the system is precisely the plane whose equation is a+b+c х — 2у + 5z %3 3 x – 2y + 5z = a х — 2у + 5z %3b х — 2у + 5z %3 с
Transcribed Image Text:1. Suppose x is an eigenvector of A corresponding to an eigenvalue 1. Show that x is an eigenvector of 51 – A. What is the corresponding eigenvalue 2? 2. Use mathematical induction to show that if 2 is an eigenvalue of an n x n matrix A, with x a corresponding eigenvector, then, for each positive integer m, 2m is an eigenvalue of A", with x a corresponding eigenvector 3. Show that if x is an eigenvector of the matrix product AB and Bx # 0, then Bx is an eigenvector of BA. ГО.4 —0.3 Lo.4 -0.5 -0.75] 4. Let A = . Explain why Ak approaches as k → o. 1.2 1.0 1.50 5. Suppose A = PDP-1, where P is 2 x 2 and D = 6 91: Let B = 51 – 3A + A². Show that B is diagonalizable by finding a suitable factorization of B. 6. If a, b, andc are distinct numbers, then the following system is inconsistent because the graphs of the equations are parallel planes. Show that the set of all least-squares solutions of the system is precisely the plane whose equation is a+b+c х — 2у + 5z %3 3 x – 2y + 5z = a х — 2у + 5z %3b х — 2у + 5z %3 с
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