Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
We have solutions for your book!
See our solution for Question 30E from Chapter 5.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 30E
Chapter:
Problem:
For each value of a in the set , compute the characteristic polynomial of A and the eigenvalues. In each case, create a graph of the characteristic polynomial for . If possible, construct all graphs on one coordinate system. Describe how the graphs reveal the changes in the eigenvalues as a changes.
Step-by-Step Solution
Given Information
We are given with a matrix: \[ A = \left[ \begin{array} { r r r } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { a } & { 25 } \end{array} \right] \] Here, a be a real value can choose from the set $\{ 32,31.9,31.8,32.1,32.2 \}$ We have to find the characteristic polynomial of A and the eigenvalues
Step-1:
Enter the matrix:
>> A= [-6 28 21; 4 -15 -12; -8 a 25];
Find the characteristic polynomial: \[ x ^ { 5 } - 11 x ^ { 4 } + 19 x ^ { 3 } + 115 x ^ { 2 } - 200 x - 500 \]
Step-2: a=32
The matrix A \[ A = \left[ \begin{array} { r r r } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 32 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ x ^ { 3 } - 4 x ^ { 2 } + 5 x - 2 \] The eigenvalues: \[ [ 2,1,1 ] \]
Step-3: a=31.9
The matrix A \[ A = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 31.9 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 3.8 x - 0.8 + x ^ { 3 } \] The eigenvalues: \[ \begin{array} { l } { [ 2.70415945787927 + 0.1,0.295840542120725 + 0.1,1.00000000000000 } \\ { \quad + 0.1 ] } \end{array} \]
Step-4: a=31.8
The matrix A \[ A : = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 31.8 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 2.6 x + 0.4 + x ^ { 3 } \] The eigenvalues: \[\left[ {1.,\,\, - 0.1278820596,\,\,3.127882060} \right]\]
Step-5: a=32.1
The matrix A \[ A : = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 32.1 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 6.2 x - 3.2 + x ^ { 3 } \] The eigenvalues: \[\left[ {1..1.500 - 0.9746794345{\rm{I}},1.500000000 + 0.9746794345{\rm{I}}} \right]\]
Step-6: a=32.2
The matrix A \[ A : = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 32.2 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 7.4 x - 4.4 + x ^ { 3 } \] The eigenvalues: \[ 1 . .1 .50000000 - 0.9746794345 \mathrm { I } , 1.500000000 + 0.9746794345 \mathrm { I } \]
Step-7
All the characteristic polynomials are plotted below:
https://imgur.com/bj3g5dR
We are given with a matrix: \[ A = \left[ \begin{array} { r r r } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { a } & { 25 } \end{array} \right] \] Here, a be a real value can choose from the set $\{ 32,31.9,31.8,32.1,32.2 \}$ We have to find the characteristic polynomial of A and the eigenvalues
Step-1:
Enter the matrix:
>> A= [-6 28 21; 4 -15 -12; -8 a 25];
Find the characteristic polynomial: \[ x ^ { 5 } - 11 x ^ { 4 } + 19 x ^ { 3 } + 115 x ^ { 2 } - 200 x - 500 \]
Step-2: a=32
The matrix A \[ A = \left[ \begin{array} { r r r } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 32 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ x ^ { 3 } - 4 x ^ { 2 } + 5 x - 2 \] The eigenvalues: \[ [ 2,1,1 ] \]
Step-3: a=31.9
The matrix A \[ A = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 31.9 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 3.8 x - 0.8 + x ^ { 3 } \] The eigenvalues: \[ \begin{array} { l } { [ 2.70415945787927 + 0.1,0.295840542120725 + 0.1,1.00000000000000 } \\ { \quad + 0.1 ] } \end{array} \]
Step-4: a=31.8
The matrix A \[ A : = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 31.8 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 2.6 x + 0.4 + x ^ { 3 } \] The eigenvalues: \[\left[ {1.,\,\, - 0.1278820596,\,\,3.127882060} \right]\]
Step-5: a=32.1
The matrix A \[ A : = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 32.1 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 6.2 x - 3.2 + x ^ { 3 } \] The eigenvalues: \[\left[ {1..1.500 - 0.9746794345{\rm{I}},1.500000000 + 0.9746794345{\rm{I}}} \right]\]
Step-6: a=32.2
The matrix A \[ A : = \left[ \begin{array} { c c c } { - 6 } & { 28 } & { 21 } \\ { 4 } & { - 15 } & { - 12 } \\ { - 8 } & { 32.2 } & { 25 } \end{array} \right] \] The characteristic polynomial: \[ - 4 x ^ { 2 } + 7.4 x - 4.4 + x ^ { 3 } \] The eigenvalues: \[ 1 . .1 .50000000 - 0.9746794345 \mathrm { I } , 1.500000000 + 0.9746794345 \mathrm { I } \]
Step-7
All the characteristic polynomials are plotted below:
https://imgur.com/bj3g5dR