If $\mathrm{A}=\left[\begin{array}{cc}0& \mathrm{x}\\ \mathrm{y}& 0\end{array}\right]$ and A³ + A = O then sum of possible values of xy is 

$\begin{array}{l}{\mathrm{A}}^3+\mathrm{A}=\mathrm{O}\\ \Rightarrow \mathrm{A}\left({\mathrm{A}}^2+\mathrm{I}\right)=\mathrm{O}\\ \Rightarrow \left|\mathrm{A}\left({\mathrm{A}}^2+\mathrm{I}\right)\right|=0\\ \Rightarrow \left|\mathrm{A}\right|=0\text{ or }\left|{\mathrm{A}}^2+\mathrm{I}\right|=0\end{array}$

If $\left|\mathrm{A}\right|=0$, then xy = 0

$\begin{array}{c}{\mathrm{A}}^2+\mathrm{I}=\left[\begin{array}{ll}0& \mathrm{x}\\ \mathrm{y}& 0\end{array}\right]\left[\begin{array}{ll}0& \mathrm{x}\\ \mathrm{y}& 0\end{array}\right]+\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}\mathrm{xy}& 0\\ 0& \mathrm{xy}\end{array}\right]+\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}\mathrm{xy}+1& 0\\ 0& \mathrm{xy}+1\end{array}\right]\\ \left|{\mathrm{A}}^2+\mathrm{I}\right|=0\\ \Rightarrow (\mathrm{xy}+1{)}^2=0\\ \Rightarrow \mathrm{xy}=-1\end{array}$

So, sum of possible values of xy is -1.