Let $q=p^n$ and let $A,B$ be two $n \times n$ matrices over $\mathbb{F}_q$ such that $AB=BA=0_n$. Let $0 \leq k \leq n$ be the nullity of a fixed $A$ (i.e. the dimension of the nullspace $\ker A$, of $A$).
A necessary and sufficient condition for $AB=0$ is that, for any $B_1 \in ImB=\{BX\:|\: X \in \mathbb{F}_q^n\}$: $AB_1=0$; hence, we must have $ImB \subseteq \ker A$. Similarly, if $BA=0$, then $Im A \subseteq \ker B$. How can I use this to prove that, for a fixed $A \in \mathscr{M}_n(\mathbb{F}_q)$, there are $q^{{k}^2}$ choices of $B$ that mutually annihilates $A$ ?
In an arXiv paper, Yifeng Huang explained this counting result by picking a certain linear map from $\mathbb{F}_q^n/ Im A\rightarrow W=\ker A$. How can I explain that ? Any help would be appreciated. Many thanks.