linear algebra - Does same characteristic polynomial and same rank imply similar?

Are two matrices with the same characteristic polynomial and the same rank necessarily similar? Where can I find the proof for such a thing?

Answer

The matrices

$$I=\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\quad\text{and}\quad A=\left(\begin{matrix}1&1\\0&1\end{matrix}\right)$$
have the same characteristic polynomial $(x-1)^2$ and are both full rank but they are not similar since the identity matrix $I$ is only similar to itself.