Let A be a 2 x 2 matrix. a) Is it true that if A is diagonalizable then A must be invertible? If so, prove it. If not, give a specific example of a matrix that is diagonalizable but not invertible (and explain why your example has these properties). b) Is it true that if A is invertible then A must be diagonalizable? If so, prove it. If not, give a specific example of a matrix that is invertible but not diagonalizable (and explain why your example has these properties).

Let A be a 2 x 2 matrix. a) Is it true that if A is diagonalizable then A must be invertible? If so, prove it. If not, give a specific example of a matrix that is diagonalizable but not invertible (and explain why your example has these properties). b) Is it true that if A is invertible then A must be diagonalizable? If so, prove it. If not, give a specific example of a matrix that is invertible but not diagonalizable (and explain why your example has these properties).

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

Let A be a 2 × 2 matrix.
a) Is it true that if A is diagonalizable then A must be invertible? If so, prove it. If not, give a specific example of a matrix
that is diagonalizable but not invertible (and explain why your example has these properties).
b) Is it true that if A is invertible then A must be diagonalizable? If so, prove it. If not, give a specific example of a matrix
that is invertible but not diagonalizable (and explain why your example has these properties).

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