If $A^3=0$ show that $A^2=0$ (using Groebner basis)

Question

Let $A$ be a $2\times 2$ matrix such that $A^3=0$. Using Groebner basis show that $A^2=0$.

Any ideas on that?

Answer 1

This is very straightforward. Write $$ A=\begin{pmatrix} a & b \cr c & d \end{pmatrix} $$ and compute $A^2$ and $A^3$. The equations for $A^3=0$ imply the ones for $A^2=0$ via a Groebner basis. Indeed, the equations for $A^3=0$ are \begin{align*} 0 & = a^3 + 2abc + bcd \\ 0 & = b(a^2 + ad + bc + d^2)\\ 0 & = c(a^2 + ad + bc + d^2)\\ 0 & = abc + 2bcd + d^3 \end{align*}

The reduced Groebner basis here gives either $c=-a^2/b, d=-a$ or $b=d=a=0$ and $c$ arbitrary. That is, $A^2=0$.