We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$. Deduce that the nilpotent matrices in $\mathcal{M}_2(\mathbb{C})$ are exactly the matrices with zero trace and zero determinant.
We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Deduce that the nilpotent matrices in $\mathcal{M}_2(\mathbb{C})$ are exactly the matrices with zero trace and zero determinant.