Past exam of the linear algebra course of the University of Oxford / 2021 / 2 / c / ii

Take the Nilpotent Jordan blockThenSuppose that a complex matrix $B$ of size three satisfied $B^3=A$. We would have $B^9=A^3=0$, so $B$ would be a nilpotent matrix. Every eigenvalue of $B$ must then be zero: if $Bv=\theta v$ with $v\ne 0$, applying $B^9$ gives ${\theta }^{9}v=0$. Its characteristic polynomial is therefore $t^3$, and the Cayley-Hamilton theorem gives $B^3=0$. This contradicts $B^3=A\ne 0$. HenceThe obstruction is that a nilpotent matrix of size three cannot have nilpotency index greater than three.