If $AB+BA=0$, then $A^2B^3=B^2A^3$? If I have a matrix $A$ and $B$ such that $AB+BA=0$
is it true that $A^2B^3=B^2A^3$?
I think that it is false.

It is false. Take
$$A=\left(\begin{array}{cc}0  & 1 \\ 1 & 0\end{array}\right),\qquad
B=\left(\begin{array}{cc}1  & 0 \\ 0 & -1\end{array}\right),$$
then we have $AB+BA=0$, but also $A^2=B^2=1$ so that $A^2B^3=B$ and $B^2A^3=A$.
