Let $S = k[x_{i,j}\mid 1\leq i\leq n, 1\leq j\leq m]$ be a polynomial ring over an arbitrary field $k$. Let $M$ be a generic $n\times m$ matrix of indeterminates in the ring $S$ where $n\leq m$. For any $k\leq n$, let $M'$ denote the matrix $M$ with column $k$ removed and let $M_k$ denote the matrix $M$ restricted to columns $1, \ldots, k$. Consider the ideals $J_1 = I_n(M')$, the ideal of maximal minors of $M'$, and $J_2 = I_k(M_k)$, the ideal of maximal minors of $M_k$. I need to show that

$$ J_1\cap J_2 = J_1\cdot J_2 $$

or, equivalently,

$$ Tor_1(S/J_1,S/J_2) = 0. $$

I feel like this must follow somehow from the fact that NO determinants in $J_1$ contain variables from column $k$, but EVERY determinant in $J_2$ contains variables from column $k$, but I am not sure how to show it.