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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.

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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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201 ces. udy fore D 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. 175 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.] 0 29. A factorization A = PDP is not unique. Demonstrate this xample 2. With D₁ = for the matrix A la [39]. 5 the information in Example 2 to find a matrix P₁ such that A = P₁D₁P₁²¹. use 30. With A and D as in Example 2, find an invertible P₂ unequal to the P in Example 2, such that A = P₂DP₂¹. 31. Construct a nonzero 2 x 2 matrix that is invertible but not diagonalizable. 32. Construct a nondiagonal 2 x 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.
Transcribed Image Text:201 ces. udy fore D 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. 175 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.] 0 29. A factorization A = PDP is not unique. Demonstrate this xample 2. With D₁ = for the matrix A la [39]. 5 the information in Example 2 to find a matrix P₁ such that A = P₁D₁P₁²¹. use 30. With A and D as in Example 2, find an invertible P₂ unequal to the P in Example 2, such that A = P₂DP₂¹. 31. Construct a nonzero 2 x 2 matrix that is invertible but not diagonalizable. 32. Construct a nondiagonal 2 x 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.
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