I am trying to prove that for any $A \in \mathbb{C}^{2\times 2}$ with $A^2\neq 0$, there exists $B \in \mathbb{C}^{2\times 2}$ with $BB=A$.

I have tried the approach of a general matrix A andB with variable entries $$B = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ $$A = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix}$$

and assuming $BB=A$, I get the equations

$$a^2+bc=\alpha$$ $$b(a+d)=\beta$$ $$c(a+d)=\gamma$$ $$dĀ²+cb=\delta$$

However, here I am stuck since I do not know whether any of those variables is $0$, so I cannot operate with those equations.

I have seen a solution on Wikipedia, however to me it seems to fall from the sky, especially the restrictions it makes.

I have also found a thread, which shows that without the restriction $A^2\neq0$ this statement is false, however I fail to see how this is the critical restriction.

Explanations, clarifications or hints on any of the things I mentioned are most welcome.