Show that if A is both diagonalizable and invertible, then so is A-¹.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 3AEXP
Related questions
Question
27please simple and short explanation
![24. A is a 3 x 3 matrix with two eigenvalues. Each eigenspace is
one-dimensional. Is A diagonalizable? Why?
25. A is a 4 x 4 matrix with three eigenvalues. One eigenspace
is one-dimensional, and one of the other eigenspaces is two-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
26. A is a 7 x 7 matrix with three eigenvalues. One eigenspace is
two-dimensional, and one of the other eigenspaces is three-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
27. Show that if A is both diagonalizable and invertible, then so
is A-¹.
28. Show that if A has n linearly independent eigenvectors, then
so does AT. [Hint: Use the Diagonalization Theorem.]
29. A factorization A = PDP-¹ is not unique. Demonstrate this
3 0
for the matrix A in Example 2. With D₁ =
[³
-[89]
5
the information in Example 2 to find a matrix P₁ such that
A = P₁D₁P₁¯¯¹.
use
30. With A and D as in Example 2, find an invertible P2 unequal
to the P in Example 2, such that A = P₂DP₂
31. Construct a nonzero 2 x 2 matrix that is invertible but not
diagonalizable.
-1
.
32. Construct a nondiagonal 2 × 2 matrix that is diagonalizable
but not invertible.
[M] Diagonalize the matrices in Exercises 33-36. Use your ma-
trix program's eigenvalue command to find the eigenvalues, and
then compute bases for the eigenspaces as in Section 5.1.
13 8 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22b1319c-242c-4915-8f56-7c035e43f773%2F012ffc49-977a-4408-b076-02947e77fd86%2Fba1nei_processed.jpeg&w=3840&q=75)
Transcribed Image Text:24. A is a 3 x 3 matrix with two eigenvalues. Each eigenspace is
one-dimensional. Is A diagonalizable? Why?
25. A is a 4 x 4 matrix with three eigenvalues. One eigenspace
is one-dimensional, and one of the other eigenspaces is two-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
26. A is a 7 x 7 matrix with three eigenvalues. One eigenspace is
two-dimensional, and one of the other eigenspaces is three-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
27. Show that if A is both diagonalizable and invertible, then so
is A-¹.
28. Show that if A has n linearly independent eigenvectors, then
so does AT. [Hint: Use the Diagonalization Theorem.]
29. A factorization A = PDP-¹ is not unique. Demonstrate this
3 0
for the matrix A in Example 2. With D₁ =
[³
-[89]
5
the information in Example 2 to find a matrix P₁ such that
A = P₁D₁P₁¯¯¹.
use
30. With A and D as in Example 2, find an invertible P2 unequal
to the P in Example 2, such that A = P₂DP₂
31. Construct a nonzero 2 x 2 matrix that is invertible but not
diagonalizable.
-1
.
32. Construct a nondiagonal 2 × 2 matrix that is diagonalizable
but not invertible.
[M] Diagonalize the matrices in Exercises 33-36. Use your ma-
trix program's eigenvalue command to find the eigenvalues, and
then compute bases for the eigenspaces as in Section 5.1.
13 8 4
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