[Math] show that a matrix a that is similar to an invertible matrix B is itself invertible. More generally, show that similar matrices have the same rank.

Question

Now I know that one of the properties of similar matrices is that they have the same rank and also that Two n x n mat. A and B are similar if B=(P^-1)AP for some invertible n x n mat. P. I looked at similar question on here but I do not understand what they are saying.

Answer 1

Product of invertible matrices is invertible.

So if $PBP^{-1}$ is invertible then so is $P^{-1}PBP^{-1}$ and so is $P^{-1}PBP^{-1}P$. The latter is equal to $B$.