Cauchy-Binet formula

Let $C$ be a $k \times m$ matrix and $D$ a $m \times k$ matrix. Then,

det (C D) = \sum_{S \subseteq {1, \dots, n}, | S | = k} det (C_{S}) \cdot det (D_{S})

where $C_{S}$ is the submatrix consisting of the $i$ th columns of $C$ for $i \in S$ and $D_{S}$ is the submatrix with the $i$ th rows of $D$ for $i \in S$
Notably, $C_{S}, D_{S}$ are both $k \times k$ matrices