Given two matrices $A \in \text{Mat}_{n\times m}$ and $B \in \text{Mat}_{m\times n}$ for $m \geq n$, I need to prove this:
$$\det(AB) = \sum\limits_I \det(A_I)\det(B_I),$$
where I passes (?) all n-element subsets of $\{1, \ldots, m\}$ and $A_I$ means $n\times n$ matrix that has columns with numbers from $I$, and $B_I$ means the same, but for rows.