Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
How do you determine which value will become λ₁ vs λ₂? Why isn’t λ₁ = 2 and λ₂ = 4 in this example?
![0
We first need the eigenvalues and eigenvectors:
=
det (A - XI₂)
=
1
1 3-X
= (A - 3)² - 1 = A²-6A +8 = (A-2) (X-4).
3-X
This gives us the distinct, real eigenvalues λ₁
=
4 and X₂ = 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd60bfbb-cf96-4101-a36e-0ab8578e2904%2F3ef9ea2c-cb2a-4bf1-8781-8ca8ea08d056%2Fxyypolj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:0
We first need the eigenvalues and eigenvectors:
=
det (A - XI₂)
=
1
1 3-X
= (A - 3)² - 1 = A²-6A +8 = (A-2) (X-4).
3-X
This gives us the distinct, real eigenvalues λ₁
=
4 and X₂ = 2.
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Follow-up Questions
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Follow-up Question
So when constructing a matrix using the eigenvectors, the order of the eigenvectors doesn't matter?
In the example they get P = ( [1, -1] , [ 1, 1]), because they put the eigenvector of eigenvalue 4 first and eigenvector of eigenvalue 2 second.
How would I know to take eigen value 4 first and not second to calculate eigenvectors?
![Based on our eigenvectors, we can construct the matrix P:
P-(1-7)
P=
We now find the inverse matrix:
1
giving us
-1 |
| 0 1
R₂
R2 1
R2-R1 R2.
R₁+R₂
6
0
P-1
=
-1 |
T
R1+R2 R1,
21
2
12 12
2,
0 1
3
(4D6969-69)
-1 3/ 1
IN
12 12
2
1
(330)
(331)
According to our theorem, P-¹AP should be a diagonal matrix whose diagonal
entries are and 2. In fact, this is the case:
(332)
(333)](https://content.bartleby.com/qna-images/question/cd60bfbb-cf96-4101-a36e-0ab8578e2904/30fc0e5b-86d9-4402-9f0c-40ddc8d85a29/1ofqz9n_thumbnail.jpeg)
Transcribed Image Text:Based on our eigenvectors, we can construct the matrix P:
P-(1-7)
P=
We now find the inverse matrix:
1
giving us
-1 |
| 0 1
R₂
R2 1
R2-R1 R2.
R₁+R₂
6
0
P-1
=
-1 |
T
R1+R2 R1,
21
2
12 12
2,
0 1
3
(4D6969-69)
-1 3/ 1
IN
12 12
2
1
(330)
(331)
According to our theorem, P-¹AP should be a diagonal matrix whose diagonal
entries are and 2. In fact, this is the case:
(332)
(333)
Solution
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