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Prove that a) If S is invertible and k is a positive integer, show that Sk is invertible and (Sk)−¹ : = (S-1)k b) Let S be an invertible matrix and let A, B be matrices such that B = S-¹AS. Show that Bk = S-¹ Ak S.

Prove that a) If S is invertible and k is a positive integer, show that Sk is invertible and (Sk)−¹ : = (S-1)k b) Let S be an invertible matrix and let A, B be matrices such that B = S-¹AS. Show that Bk = S-¹ Ak S.

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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**Prove that:** a) If \( S \) is invertible and \( k \) is a positive integer, show that \( S^k \) is invertible and \((S^k)^{-1} = (S^{-1})^k\). b) Let \( S \) be an invertible matrix and let \( A, B \) be matrices such that \( B = S^{-1}AS \). Show that \( B^k = S^{-1}A^kS \).
Transcribed Image Text:**Prove that:** a) If \( S \) is invertible and \( k \) is a positive integer, show that \( S^k \) is invertible and \((S^k)^{-1} = (S^{-1})^k\). b) Let \( S \) be an invertible matrix and let \( A, B \) be matrices such that \( B = S^{-1}AS \). Show that \( B^k = S^{-1}A^kS \).
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