Let A be a matrix that satisfies ?^2 =? where ? is the identity matrix. a. Find a 2x2 example of such a matrix that is not 1 0 0 1 or -1 0 0 -1 b. Prove that, in general (i.e, any size), any matrix that satisfies ?^2 =? must be invertible. c. Explain why it must be the case that ?^−1 =?.
Let A be a matrix that satisfies ?^2 =? where ? is the identity matrix. a. Find a 2x2 example of such a matrix that is not 1 0 0 1 or -1 0 0 -1 b. Prove that, in general (i.e, any size), any matrix that satisfies ?^2 =? must be invertible. c. Explain why it must be the case that ?^−1 =?.
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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Let A be a matrix that satisfies ?^2 =? where ? is the identity matrix.
a. Find a 2x2 example of such a matrix that is not
| 1 | 0 |
| 0 | 1 |
or
| -1 | 0 |
| 0 | -1 |
b. Prove that, in general (i.e, any size), any matrix that satisfies ?^2 =? must be invertible.
c. Explain why it must be the case that ?^−1 =?.
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