All $n$ x $n$ matrices with characteristic polynomial $f(x)$ are similar iff $f(x)$ has no repeated roots in F[x].
My attempt: If f has no repeated matrices, all such matrices are diagonlisable hence similar. The other direction I haven't been able to do. Here's my line of thinking: if two matrices are similar, they have the same rational canonical form hence the same invariant factors(?). Thus they have the same minimal polynomial, but I am not sure why the characteristic and minimal polynomials are equal.