True or false? If two matrices $A,B$ are nilpotent and have the same Jordan normal form, $A+B$ is also nilpotent.

Question

I do know that $(n \times n)$ nilpotent matrices have the minimal polynomial $x^k$ for some positive integer $k ≤ n$.

I also do know that having the same Jordan normal form means they have the same minimal polynomial.

Any hints are welcome, I strongly suspect this statement to be false.

Answer 1

False: consider $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$.

Answer 2

Following on Daniel's answer this would imply every symmetric matrix with $0$ on the diagonal is nilpotent.

This is known to be false to anyone who does spectral graph theory.