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 $A^2=0$

suppose $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, then $A^2=0$ makes a four equation group of four unknown numbers($a,b,c,d$) like:
$$a^2+bc = 0\\ab+bd = 0\\ac+cd = 0\\bc+d^2 = 0$$
solve these equations, we have two possible solutions:
$$a=0, b=0,c\in R, d=0$$
or
$$a\in R,b\in R,c = -a^2/b,d=-a.$$
So the forms of matrix are:
$$\begin{bmatrix}a&b\\-a^2/b&-a\end{bmatrix}$$
or
$$\begin{bmatrix}0&0\\c&0\end{bmatrix}$$
