General form for $A^2=0$ where $A$ is a $2\times2$ matrix Find the general form for all the $2\times2$ matrices over $\mathbb{R}$ that $A^2=0$

Either $A=0$, or $A\neq 0$ and there exists $x\in\mathbb R^2$ such that $\{x,Ax\}$ is a basis of $\mathbb R^2$. We thus have the general form of $A$ up to a change of basis.
