Explanation Given: The given matrix is: A = [ 2 1 3 0 2 − 1 0 0 2 ] Formula used: The characteristics equation is: | A − r I | = 0 Here, r is the eigenvalues. The Cayley-Hamilton theorem: It states that a matrix satisfy its own characteristics equation, that is p ( A ) = 0 . Proof: Consider the given matrix: A = [ 2 1 3 0 2 − 1 0 0 2 ] The characteristics equation is: | [ 2 1 3 0 2 − 1 0 0 2 ] − r [ 1 0 0 0 1 0 0 0 1 ] | = 0 | [ 2 1 3 0 2 − 1 To determine (b) The matrix e A t for the given matrix.

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

See similar textbooks

Videos

Question

Chapter 9.8, Problem 4E

To determine

(a)

To show:

The given matrix satisfies (A−rI)k=0 for some r and some positive integer k.

To determine

(b)

The matrix eAt for the given matrix.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.8, Problem 3E

Chapter 9.8, Problem 5E

Students have asked these similar questions

25 Topology: Properties of Compact Spaces Task: Refer to Question 25 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC44OAZF/view?usp=sharing

4 6 Spectral Theory: Eigenfunctions and Eigenvalues Task: Refer to Question 46 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing 47 Category Theory: Functors and Natural Transformations Task: Refer to Question 47 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing

= Consider the vector space R² with the standard basis {e1 {v1 = (1, 1), V2 = = (1, 0), e2 = (0, 1)} and a new basis new basis. (1,-1)}. a) Find the change of basis matrix from the standard basis to the b) Express the vector w = (2, 3) in the new basis {V1, V2}.

Chapter 9 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

(a) To show: The given matrix satisfies ( A − r I ) k = 0 for some r and some positive integer k.

Solution Summary: The author explains the Cayley-Hamilton theorem that a matrix satisfy its own characteristics equation, that is p(A)=0.

Videos

(a)

To show:

The given matrix satisfies (A−rI)k=0 for some r and some positive integer k.

(b)

The matrix eAt for the given matrix.

Want to see the full answer?

Chapter 9 Solutions

(a) To show: The given matrix satisfies ( A − r I ) k = 0 for some r and some positive integer k.

Solution Summary: The author explains the Cayley-Hamilton theorem that a matrix satisfy its own characteristics equation, that is p(A)=0.

Videos

(a) To show: The given matrix satisfies (A−rI)k=0 for some r and some positive integer k.

(b) The matrix eAt for the given matrix.

Want to see the full answer?

Chapter 9 Solutions

(a)

To show:

The given matrix satisfies (A−rI)k=0 for some r and some positive integer k.

(b)

The matrix eAt for the given matrix.