Similar matrices Q

I'm a little rusty with my linear algebra, and was wondering how to prove that two matrices are similar. The question is:

Two matrices A and B are said to be similar if there exists an invertible matrix S such that . Prove that A ~ A.

So they want me to prove that , if I'm not mistaken. How do I show this? If I recall correctly, matrices are not commutative, so it's not a matter of cancelling out the S terms. Any guidance would be great!

So I always treat S as an identity matrix?

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Originally Posted by Glitch

So I always treat S as an identity matrix?

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Not always; only in this case to show that A~A.

Thanks. So say that I was proving A ~ B instead, how would the process differ?

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Originally Posted by Glitch

Thanks. So say that I was proving A ~ B instead, how would the process differ?

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The matrix S would depend on what A and B actually were. If A and B were different (yet similar!), we would get an S which was different from I.

Ahh ok, thanks again. :)