Proof/Counterexample: Any two 6x6 matrices are similar of they have the same rank and same minimal polynomial

Question

I am asked to prove or provide a counterexample for this claim:

Any two $6\times6$ matrices are similar of they have the same rank and same minimal polynomial.

I know two matrices are similar if they have the same Jordan form. I have a hunch that they are similar always in this case and no counterexample would exist but I don't know how to prove it.

Thanks in advance.

Edit: I found this answer which is along the similar lines but how to prove that the Jordan forms will be same?

Answer 1

Are $$A=\begin{pmatrix}1&1&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1 \end{pmatrix}$$ and $$B=\begin{pmatrix}1&1&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1 \end{pmatrix}$$ similar? Both have rank $6$ and minimal polynomial $X^2-2X+1$. (This works already with $4\times 4$ matrices)