Explanation Given: The matrix is, A = [ 3 1 0 − 1 5 0 0 0 4 ] Approach: For a given n × n matrix A , the polynomial p ( λ ) defined by p ( λ ) = det ( A − λ I ) is called the characteristic polynomial of A , and the equation p ( λ ) = 0 is called the characteristic equation of A . The polynomial of the form, p ( λ ) = ( − 1 ) n ( λ − λ 1 ) m 1 ( λ − λ 2 ) m 2 ⋯ ( λ − λ k ) m k Thus, associated with each eigenvalue λ i is a number m i , called the multiplicity of λ i . Let A be an n × n matrix. For a given eigenvalue λ i , let E i denotes the set of all vectors v satisfying A v = λ i v . Then E i is called the eigenspace of a A corresponding to the eigenvalue λ i . Thus, E i is the solution set to the linear system ( A − λ i I ) v = 0 . If n i denotes the dimension of E i , then 1 ≤ n i ≤ m i for each i . In words, the dimension of the eigenspace corresponding to λ i is at most the multiplicity of λ i . An n × n matrix A that has n linearly independent eigenvectors is called nondefective. In such a case, we say that A has a complete set of eigenvectors. If A has less than n linearly independent eigenvectors, it is called defective. Calculation: Calculate the eigenvalues for the matrix A . The characteristic polynomial of A can be written as, p ( λ ) = | 3 − λ 1 0 − 1 5 − λ 0 0 0 4 − λ | = ( 3 − λ ) [ ( 5 − λ ) ( 4 − λ ) ] − 1 ( − 4 + λ ) = ( 4 − λ ) ( λ 2 − 8 λ + 16 ) = ( 4 − λ ) ( λ − 4 ) 2 The characteristic equation of A is, ( 4 − λ ) ( λ − 4 ) 2 = 0 λ = 4 , 4 , 4 The eigenvalues for the matrix A are λ 1 = 4

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Question

Chapter 7.2, Problem 8P

To determine

The multiplicity of each eigenvalue, basis for each eigenspace, dimension of each eigenspace of the given matrix A and whether the matrix is defective or nondefective.

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Chapter 7.2, Problem 7P

Chapter 7.2, Problem 9P

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MODELING REAL LIFE Your checking account has a constant balance of $500. Let the function $m$ represent the balance of your savings account after $t$ years. The table shows the total balance of the accounts over time. Year, $t$ Total balance 0 1 2 3 4 5 $2500 $2540 $2580.80 $2622.42 $2664.86 $2708.16 a. Write a function $B$ that represents the total balance after $t$ years. Round values to the nearest hundredth, if necessary. $B\left(t\right)=$ Question 2 b. Find $B\left(8\right)$ . About $ a Question 3 Interpret $B\left(8\right)$ . b represents the total balance checking and saving accounts after 8 years the balance would be 16 / 10000 Word Limit16 words written of 10000 allowed Question 4 c. Compare the savings account to the account, You deposit $9000 in a savings account that earns 3.6% annual interest compounded monthly. A = 11998.70 SINCE 9000 is the principal ( 1+0.036/12)12 times 8 gives me aproxtimately 1997 14 / 10000 Word Limit14 words written of 10000 allowed Skip to…

Chapter 7 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 7.1 - Prob. 1P Ch. 7.1 - Prob. 2P Ch. 7.1 - Prob. 3P Ch. 7.1 - Prob. 4P Ch. 7.1 - Prob. 5P Ch. 7.1 - Given that v1=(2,1) and v2=(1,1) are eigenvectors...Ch. 7.1 - Prob. 7P Ch. 7.1 - Prob. 8P Ch. 7.1 - Prob. 9P Ch. 7.1 - Prob. 11P

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