Binary rank of the adjacency matrix

The 2-rank of the adjacency matrix of a graph is always even. There are various proofs for this; see, for example, Chapter 8 in [1], where this is shown as part of a larger theorem. However, while teaching a class I noticed the following simple proof.

Claim: Let G be a simple n-vertex graph. Then the 2-rank of A(G) is even.

Proof: If $n\leq 2$ or G is empty: ✓. Proceed by induction on n. Order the vertices $v_1,v_2,\dots,v_n$ so that $v_1$ and $v_2$ are adjacent. In A(G), subtract column 1 from every column associated to a vertex in $N(v_2)-v_1$ and column 2 from every column associated to a vertex in $N(v_1)-v_2$ . Do the same with the rows. The resulting matrix is a block matrix whose blocks are $\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ and A’, where A’ is the adjacency matrix of an (n-2)-vertex graph. □