For a $2 \times 2$ matrix $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$, prove that $A^{2}=0$ if and only if $a=-d$ and $b c=-a^{2}$. Prove that $A^{3}=0$ if and only if $A^{2}=0$.

[Hint: it is easy to check that $\left.A^{2}-(a+d) A+(a d-b c) I=0 .\right]$