We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$.
Calculate the trace and rank of $A$. Deduce, without any calculation, the characteristic polynomial of $A$. Show that $A$ is nilpotent and give its nilpotency index.