Relationship Between Similar Matrices A and B and their Inverses

Question

This question came up in my linear algebra class, and I'm having trouble answering it. I found this same question on Math Stack Exchange, but it was never properly answered (although marked as answered):

Show that for invertible matrices $A$ and $B$, if $A$ is similar to $B$ ($B = P^{-1}AP$), then $A^{-1}$ is similar to $B^{-1}$.

Thanks for your help!

Answer 1

$B=P^{-1}AP$

$B^{-1}=(P^{-1}AP)^{-1}=P^{-1}A^{-1}(P^{-1})^{-1}=P^{-1}A^{-1}P$

$PB^{-1}P^{-1}=A^{-1}$

Answer 2

Well, since

$B = P^{-1}AP$

and

$(P^{-1}AP)(P^{-1}A^{-1}P) = I$,

we see that

$B^{-1} = P^{-1}A^{-1}P$