Explanation Given: The given matrix is: A = [ 1 − 1 1 3 ] 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 = [ 1 − 1 1 3 ] The characteristics equation is: | [ 1 − 1 1 3 ] − r [ 1 0 0 1 ] | = 0 | [ 1 − 1 1 3 ] − [ r 0 0 r ] | = 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

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Chapter 9.8, Problem 2E

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.

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Chapter 9.8, Problem 1E

Chapter 9.8, Problem 3E

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Chapter 9 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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(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, which states that a matrix satisfy its own characteristics equation, that is p(A).

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, which states that a matrix satisfy its own characteristics equation, that is p(A).

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.