True/False: If the statement is false, justify why it is false. (a) If a matrix is invertible, then it is diagonalizable. 1 2 -1 2 04-3 00 3 2 2 00 0 -2, (b) 2 is an eigenvalue of A = (c) If Ax = Xx for some vector x, then A is an eigenvalue of A. (d) If λ + 2 is a factor of the characteristic polynomial of A, then 2 is an eigenvalue of A. (e) In order for an n x n matrix A to be diagonalizable, A must has n distinct eigenvalues.
True/False: If the statement is false, justify why it is false. (a) If a matrix is invertible, then it is diagonalizable. 1 2 -1 2 04-3 00 3 2 2 00 0 -2, (b) 2 is an eigenvalue of A = (c) If Ax = Xx for some vector x, then A is an eigenvalue of A. (d) If λ + 2 is a factor of the characteristic polynomial of A, then 2 is an eigenvalue of A. (e) In order for an n x n matrix A to be diagonalizable, A must has n distinct eigenvalues.
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:**True/False: If the statement is false, justify why it is false.**
(a) If a matrix is invertible, then it is diagonalizable.
(b) 2 is an eigenvalue of \( A = \begin{pmatrix} 1 & 2 & -1 & 2 \\ 0 & 4 & -3 & 2 \\ 0 & 0 & 3 & 2 \\ 0 & 0 & 0 & -2 \end{pmatrix} \).
(c) If \( Ax = \lambda x \) for some vector \( x \), then \( \lambda \) is an eigenvalue of \( A \).
(d) If \( \lambda + 2 \) is a factor of the characteristic polynomial of \( A \), then 2 is an eigenvalue of \( A \).
(e) In order for an \( n \times n \) matrix \( A \) to be diagonalizable, \( A \) must have \( n \) distinct eigenvalues.
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