Two matrices $A, B$ satisfying in characteristic polynomial of $B$ and $A$, respectively.

Question

Let matrices $A,B\in M_n(\mathbb{C})$ such that $A$ satisfy in characteristic polynomial of $B$, and $B$ satisfy in characteristic polynomial of $A$.

Can we say that:

$A$ is diagonalizable if and only if $B$ be diagonalizable.

Answer 1

It is not true for the characteristic polynomial. For example take the zero matrix of order 2 ,and take the 2×2 matrix with zero's everywhere except the top right corner. They have the same characteristic polynomial X^2 and therefore the condition holds, but the first is diagonizable and the second isnt.