The Stacks project

  • given matrices $A$ and $B$ in a ring $R$ of sizes $m \times n$ and $n \times m$ we have $\det (AB) = \sum \det (A_ S)\det ({}_ SB)$ in $R$ where the sum is over subsets $S \subset \{ 1, \ldots , n\} $ of size $m$ and $A_ S$ is the $m \times m$ submatrix of $A$ with columns corresponding to $S$ and ${}_ SB$ is the $m \times m$ submatrix of $B$ with rows corresponding to $S$,

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