We know that $A\in M_{m n}(\mathbb R)\land B\in M_{n m}(\mathbb R)$ where $2\leq n\leq m\leq 3$.
If $4(AB)^3+3(AB)^2+2(AB)+I_m=0_m$, which one of the following holds?:
$(a)$$m=2$
$(b)$ $m=3$
$(c)$ $n=2$
$(d)$ $m=n$
$(e)$ $(AB)^4=I_m$
$(f)$ $\det(AB)=0$
$n=2$ or $m=n$ don't seem to change anything in the equality.
We can rewrite the given equality as $AB(-4(AB)^2-3(AB)-2I_m)=I_m$, which means that AB is nonsingular, so $f$ is ruled out. The most logical answer to me seems to be e), but I didn't manage to prove it. Any help is greatly appreciated!