1. True/False: If the statement is false, justify why it is false. (a) For n x n matrices A and B, det(AB) + det(A)det(B). (b) If a matrix is invertible, then it is diagonalizable. 1 2 -1 0 4 -3 0 0 5 0 0 0 -5 2 5 (c) 5 is an eigenvector of A = 1 (d) If Ax = Ax for some vector x, then A is an eigenvalue of A. (e) If A + 3 is a factor of the characteristic polynomial of A, then –3 is an eigenvalue of A. (f) In order for an n xn matrix A to be diagonalizable, A must has n distinct eigenvalues. 2. If A is a 3 x 3 matrix, prove that the determinant of A is equal to the determinant of A". 3. If the eigenvalues of A are 2 and -1, with multiplicities 2 and 3, respectively, find the characteristic equation of A. 4. Find a 4 x 4 matrix A such that the eigenvalues of A are -2, 3, and 0.

1. True/False: If the statement is false, justify why it is false. (a) For n x n matrices A and B, det(AB) + det(A)det(B). (b) If a matrix is invertible, then it is diagonalizable. 1 2 -1 0 4 -3 0 0 5 0 0 0 -5 2 5 (c) 5 is an eigenvector of A = 1 (d) If Ax = Ax for some vector x, then A is an eigenvalue of A. (e) If A + 3 is a factor of the characteristic polynomial of A, then –3 is an eigenvalue of A. (f) In order for an n xn matrix A to be diagonalizable, A must has n distinct eigenvalues. 2. If A is a 3 x 3 matrix, prove that the determinant of A is equal to the determinant of A". 3. If the eigenvalues of A are 2 and -1, with multiplicities 2 and 3, respectively, find the characteristic equation of A. 4. Find a 4 x 4 matrix A such that the eigenvalues of A are -2, 3, and 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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Question

1. True/False: If the statement is false, justify why it is false.
(a) For n x n matrices A and B, det(AB) + det(A)det(B).
(b) If a matrix is invertible, then it is diagonalizable.
1 2 -1
0 4 -3 5
0 0
0 0
2
(c) 5 is an eigenvector of A =
1
-5
(d) If Ax = Ax for some vector x, then A is an eigenvalue of A.
(e) If A+3 is a factor of the characteristic polynomial of A, then –3 is an eigenvalue of A.
(f) In order for an n xn matrix A to be diagonalizable, A must hasn distinct eigenvalues.
2. If A is a 3 x 3 matrix, prove that the determinant of A is equal to the determinant of A".
3. If the eigenvalues of A are 2 and -1, with multiplicities 2 and 3, respectively, find the
characteristic equation of A.
4. Find a 4 x 4 matrix A such that the eigenvalues of A are -2, 3, and 0.

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