Let $\mathcal{L}_n$ be the set of all $n \times n$ nonsingular matrices with elements over $\mathbb{F}_2 = \{0, 1\}$. Is it possible to find a sequence
$$ A_1, A_2, \dots, A_N $$
where $A_i \in \mathcal{L}_n$ and $N = |\mathcal{L}_n|$, such that $A_i \neq A_j$ for all pairs $1 \le i < j \le N$ and $A_{i+1}$ is obtained from $A_i$ by adding some row to another one, for all $n$? Addition is modulo-2 and consider $N+1 = 1$, i.e., the sequence describes a cycle.