Let's call a set of matrices $\mathcal{A} = \{A_1, \ldots, A_k\}$ nilpotent if there is some $T$ such that the product of every $T$ matrices from $\mathcal{A}$ equals zero.
Then we have the following theorem: A set of $n \times n$ matrices $\mathcal{A}$ is nilpotent if and only if the product of every $n$ matrices from this set equals zero.
Is there a simple/natural proof of this? This statement is Proposition 2.1 here. I can't follow the proof in that document, and in any case it is an induction on dimension, whereas it feels like this statement might have a better/conceptual argument.