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## Homework Statement

Given two matrices, [tex] A [/tex] and [tex] B.[/tex] Is the following statement true?

[tex] A [/tex] is similar to [tex] B [/tex] [tex] \Longleftrightarrow [/tex] [tex] A^k [/tex] is similar to [tex] B^k. [/tex]

## Homework Equations

By definition, A and B are similar if there exists an invertible matrix P such that B = P

^{-1}AP.

## The Attempt at a Solution

Clearly, the [tex] \Rightarrow [/tex] portion of the statement holds. For example, B

^{2}= (P

^{-1}AP)(P

^{-1}AP) = P

^{-1}A(PP

^{-1})AP = P

^{-1}A

^{2}P.

However, I am not certain about the statement in the reverse direction. I haven't spent a terrible amount of time on it, but I can't think of any counterexamples straight off the top of my head.

Any hints or suggestions? Thanks for your time.