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(a) Suppose A is an invertible matrix. If A is an eigenvalue of A, show that is an eigenvalue of A¹.

(a) Suppose A is an invertible matrix. If A is an eigenvalue of A, show that is an eigenvalue of A¹.

Advanced Engineering Mathematics
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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Problem 5. (a) Suppose A is an invertible matrix. If A is an eigenvalue of A, show that is an eigenvalue of A¹. (b) Show that if A is invertible and diagonalizable, then A¹ is diagonalizable (hint: use part (a)) (c) If A is an eigenvalue of A, show that A² - 4 is an eigenvalue of A² - 41, where I is the identity matrix. (d) Suppose A has a zero eigenvalue. Explain why A is not invertible (i.e. its inverse does not exist).
Transcribed Image Text:Problem 5. (a) Suppose A is an invertible matrix. If A is an eigenvalue of A, show that is an eigenvalue of A¹. (b) Show that if A is invertible and diagonalizable, then A¹ is diagonalizable (hint: use part (a)) (c) If A is an eigenvalue of A, show that A² - 4 is an eigenvalue of A² - 41, where I is the identity matrix. (d) Suppose A has a zero eigenvalue. Explain why A is not invertible (i.e. its inverse does not exist).
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