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Problem 0.5 Let A be a 2x2 matrix which has real eigenvalues X1, A2, counted with multiplicity. Determine whether the following statements are true or false, without justifications. (1) If X₁ X2, then A is diagonalizable.

Problem 0.5 Let A be a 2x2 matrix which has real eigenvalues X1, A2, counted with multiplicity. Determine whether the following statements are true or false, without justifications. (1) If X₁ X2, then A is diagonalizable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 0.5** Let \( A \) be a \( 2 \times 2 \) matrix which has real eigenvalues \( \lambda_1, \lambda_2 \), counted with multiplicity. Determine whether the following statements are true or false, *without* justifications. 1. If \( \lambda_1 \neq \lambda_2 \), then \( A \) is diagonalizable.
Transcribed Image Text:**Problem 0.5** Let \( A \) be a \( 2 \times 2 \) matrix which has real eigenvalues \( \lambda_1, \lambda_2 \), counted with multiplicity. Determine whether the following statements are true or false, *without* justifications. 1. If \( \lambda_1 \neq \lambda_2 \), then \( A \) is diagonalizable.
2. If \( \lambda_1 = \lambda_2 \), then \( A \) is not diagonalizable. 3. \( A \) is always diagonalizable. 4. \( \det A = \lambda_1 \lambda_2 \). 5. (Optional) Now suppose \( A \) is a \( 3 \times 3 \) matrix which has three distinct real eigenvalues \( \lambda_1, \lambda_2, \lambda_3 \). Then \( A \) is diagonalizable.
Transcribed Image Text:2. If \( \lambda_1 = \lambda_2 \), then \( A \) is not diagonalizable. 3. \( A \) is always diagonalizable. 4. \( \det A = \lambda_1 \lambda_2 \). 5. (Optional) Now suppose \( A \) is a \( 3 \times 3 \) matrix which has three distinct real eigenvalues \( \lambda_1, \lambda_2, \lambda_3 \). Then \( A \) is diagonalizable.
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