Suppose I have two $m\times n$ matrices $A$ and $B$ such that an $m\times m$ submatrix of $A$ is invertible if and only if the corresponding $m \times m$ submatrix of $B$ is. Now let's say I append a row at the bottom of $A$, and we can denote this new $(m+1)\times n$ matrix as $A_2$. Can I always find a row to append onto the bottom of $B$ such that an $(m+1)\times (m+1)$ submatrix of $A_2$ is invertible if and only if the corresponding $(m+1)\times (m+1)$ submatrix of $B_2$ is? $B_2$ is the new matrix obtained from adding a row onto $B$.
I suspect this is dependent upon which field the matrices have entries in, but I've made no luck in this problem so far. I've tried thinking about it by trying to compute determinants via Laplace expansion, using 3-term Plucker relations, and using determinant formulas for block matrices, but that hasn't helped.
Update: I'm not sure if it helps, but we also can think of it as though we have a system of $\binom{n}{m}$ linear (or polynomial?) equations, $n$ variables in total but each equation having $m$ variables in them and $n!$ terms, with some equations equaling 0 and others aren't. So perhaps homotopy continuation might be helpful?
