A is a 5 x 5 matrix with two eigenvalues. One eigenspace is three-dimensional, and the other eigenspace is two- dimensional. Is A diagonalizable? Why?
A is a 5 x 5 matrix with two eigenvalues. One eigenspace is three-dimensional, and the other eigenspace is two- dimensional. Is A diagonalizable? Why?
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
Problem 83E
Related questions
Question
23 please simple and short ex
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1
17.
19.
5
4
0
0
0
0
4
0
-3
3
0
0
0
5
9
1 -2
0
2
0120
18.
20.
-7
you try these exercises.)
In Exercises 21 and 22, A, B, P, and D are nxn matrices.
Mark each statement True or False. Justify each answer. (Study
Theorems 5 and 6 and the examples in this section carefully before
-16
4
6 13 -2
12
16
4
0400
5
0000
0
2
0002
21. a.
and some invertible matrix P.
A is diagonalizable if A = PDP-¹ for some matrix D
nalizable.
b. If R" has a basis of eigenvectors of A, then A is diago-
c. A is diagonalizable if and only if A has n eigenvalues,
counting multiplicities.
d. If A is diagonalizable, then A is invertible.
is A-1.
28. Show that if A has n
so does AT. [Hint: U
29. A factorization A =
for the matrix A in
c. If AP = PD, with D diagonal, then the nonzero columns
of P must be eigenvectors of A.
d. If A is invertible, then A is diagonalizable.
the information in
A = P₁D₁P₁¹.
30. With A and D as i
to the P in Exam
A is both
31. Construct a nonz
diagonalizable.
32. Construct a non
but not invertibl
23. A is a 5 x 5 matrix with two eigenvalues. One eigenspace
is three-dimensional, and the other eigenspace is two-
dimensional. Is A diagonalizable? Why?
22. a. A is diagonalizable if A has n eigenvectors.
V. If A is diagonalizable, then A has n distinct eigenvalues. 35.
[M] Diagonalize the
trix program's eigen
then compute bases
33.
36.
-6 4
-3
0
-2
11 -6
-3 5
-8 12
6
-18
1
8
4
0 1
6 12
92
15 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22b1319c-242c-4915-8f56-7c035e43f773%2F49116ea0-5cd4-476d-b17c-79ed3350eec2%2Fv1f58c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1
1
17.
19.
5
4
0
0
0
0
4
0
-3
3
0
0
0
5
9
1 -2
0
2
0120
18.
20.
-7
you try these exercises.)
In Exercises 21 and 22, A, B, P, and D are nxn matrices.
Mark each statement True or False. Justify each answer. (Study
Theorems 5 and 6 and the examples in this section carefully before
-16
4
6 13 -2
12
16
4
0400
5
0000
0
2
0002
21. a.
and some invertible matrix P.
A is diagonalizable if A = PDP-¹ for some matrix D
nalizable.
b. If R" has a basis of eigenvectors of A, then A is diago-
c. A is diagonalizable if and only if A has n eigenvalues,
counting multiplicities.
d. If A is diagonalizable, then A is invertible.
is A-1.
28. Show that if A has n
so does AT. [Hint: U
29. A factorization A =
for the matrix A in
c. If AP = PD, with D diagonal, then the nonzero columns
of P must be eigenvectors of A.
d. If A is invertible, then A is diagonalizable.
the information in
A = P₁D₁P₁¹.
30. With A and D as i
to the P in Exam
A is both
31. Construct a nonz
diagonalizable.
32. Construct a non
but not invertibl
23. A is a 5 x 5 matrix with two eigenvalues. One eigenspace
is three-dimensional, and the other eigenspace is two-
dimensional. Is A diagonalizable? Why?
22. a. A is diagonalizable if A has n eigenvectors.
V. If A is diagonalizable, then A has n distinct eigenvalues. 35.
[M] Diagonalize the
trix program's eigen
then compute bases
33.
36.
-6 4
-3
0
-2
11 -6
-3 5
-8 12
6
-18
1
8
4
0 1
6 12
92
15 2
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