Consider a matrix $\tilde A$ of size $(N+2)\times (N+2)$ whose coefficients are either $0$ or $1$.
Denote $A$ the submatrix containing columns from 2 to $N$ of $\tilde A$ and line from $2$ to $N$ of $\tilde A$. In short $A=\tilde A(2...N,2...N)$.
Define the following elementary operator: $$b_i^j(A) = A + E_{i,j}+ E_{i,j+1}+ E_{i+1,j}+ E_{i-1,j}+ E_{i,j-1} \quad \text{mod}\; [2]$$ for all $i,j$ of $\mathbf{\{1,\ldots, N\}}$ where $E_{a,b}$ is the elementary matrix (which has 0 everywhere except for the coefficient in position $(a,b)$).
Is it possible to transform any submatrix $M\in M_{N,N}(\{0,1\})$ of a matrix $\tilde M\in M_{N+2,N+2}(\{0,1\})$ to the null matrix of $M_{N,N}(\{0,1\})$ using these operations $b_i^j$ ?
Remark: Assuming the existence of such algorithm and applying it to the matrix $\tilde M$, we obtain a matrix denoted as $\tilde M_{\infty}$. We do not care about the coefficients of the first and last columns/lines of $\tilde M_{\infty}$, we only care that the submatrix $M_{\infty}=\tilde M_{\infty}(2...N,2...N)$ is null.