Consider the NxN-dimensional anti-symmetric matrix A, with N is odd, and the matrix B=A². a) Prove that the determinant of A is equal to 0. b) Which of the following statements is true? 1. The diagonal elements of A are equal to 0. 2. The diagonal elements of B are equal to 0. 3. The trace of A is equal to 0. 4. The trace of B is equal to 0. 5. The determinant of B is equal to the determinant of A. 6. The rank of B is equal to the rank of A. 7. All eigenvalues of B are equal to 0. 8. At least one eigenvalue of B is equal to 0. 9. Exactly one eigenvalue of B is equal to 0. 10. All eigenvalues of B are positive or equal to 0.
Consider the NxN-dimensional anti-symmetric matrix A, with N is odd, and the matrix B=A². a) Prove that the determinant of A is equal to 0. b) Which of the following statements is true? 1. The diagonal elements of A are equal to 0. 2. The diagonal elements of B are equal to 0. 3. The trace of A is equal to 0. 4. The trace of B is equal to 0. 5. The determinant of B is equal to the determinant of A. 6. The rank of B is equal to the rank of A. 7. All eigenvalues of B are equal to 0. 8. At least one eigenvalue of B is equal to 0. 9. Exactly one eigenvalue of B is equal to 0. 10. All eigenvalues of B are positive or equal to 0.
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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Transcribed Image Text:Consider the NxN-dimensional anti-symmetric matrix A, with N is odd, and the matrix B=A².
a) Prove that the determinant of A is equal to 0.
b) Which of the following statements is true?
1. The diagonal elements of A are equal to 0.
2. The diagonal elements of B are equal to 0.
3. The trace of A is equal to 0.
4. The trace of B is equal to 0.
5. The determinant of B is equal to the determinant of A.
6. The rank of B is equal to the rank of A.
7.
All eigenvalues of B are equal to 0.
8. At least one eigenvalue of B is equal to 0.
9. Exactly one eigenvalue of B is equal to 0.
10. All eigenvalues of B are positive or equal to 0.
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