If $A,B\in M_n(\mathbb{F})$ are $n-1$ nilpotent, prove they are similar.
Can I say that, since their minimal polynomial is $X^{n-1}$ they are similar?
I know that If $A,B$ are similar, they have the same minimal polynomial. Does it work the other way?
To answer your question about minimal polynomial and similar matrices, if you take the two following matrices:
$$\begin{pmatrix} \lambda & 1 &0 &0\\ 0 & \lambda &0 &0\\ 0 & 0 &\lambda &1\\ 0 & 0 &0 &\lambda\\ \end{pmatrix} $$ $$\begin{pmatrix} \lambda & 1 &0 &0\\ 0 & \lambda &0 &0\\ 0 & 0 &\lambda &0\\ 0 & 0 &0 &\lambda\\ \end{pmatrix} $$
These two matrices have the same minimal polynomial, $(X-\lambda)^2$, but they are not similar. Even if your matrix is nilpotent in which case $\lambda =0.$