Give an example that shows there are $4\times4$ matrices over $\mathbb{Q}$

Question

Can someone help me solve this problem?

Give an example that shows there are $4\times4$ matrices over $\mathbb{Q}$ with the same characteristic and minimal polynomials that are not similar.

Answer 1

Consider $A=\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\\\end{pmatrix}$ and $B=\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}$.

Check that both have same minimal polynomial as well as characteristic polynomial. But since $rank(A)=2$ and $rank(B)=1$ they are not similer (why? See the problem on MSE).