Please give an example,such that:
$\displaystyle A,B \in M_{n}(\mathbb{F})$
and $\displaystyle AB$ has the same eigenpolynomial as $\displaystyle BA$
but they don't have the same minimal polynomial
In order to have the same minimal polynomial, the matrices must have the same eigenvalues. In order not to have the same characteristic polynomial, those eigenvalues must not have the same multiplicity. Try two 3 by 3 matrices with two distinct eigenvalues, one of them of multiplicity 2.
(And you can make this problem really simple by using diagonal matrices.)