Solving systems of linear equations through zero forcing set and application to lights out

Let $\mathbb{F}$ be any field, we consider solving $Ax=b$ repeatedly for a matrix $A\in {\mathbb{F}}^{n\times n}$ of $m$ non-zero elements, and multiple different $b\in {\mathbb{F}}^n$. If we are given a zero forcing set of $A$ of size $k$, we can then build a data structure in $O(mk)$ time, such that each instance of $Ax=b$ can be solved in $O(k^2+m)$ time. As an application, we show how the lights out game in an $n\times n$ grid is solved in $O(n^3)$ time, and then improve the running time to $O(n^{\omega }\log n)$ by exploiting the repeated structure in grids.