Cauchy–Binet formula

For matrices $A\in {\mathbb{R}}^{r\times m}$ and $B\in {\mathbb{R}}^{m\times r}$ with $r\le m$,where the indices of $S$ are taken in increasing order. To prove it, write the $j$th column of $AB$ as ${\sum }_q{B}_{qj}{A}_{:,q}$ and expand its determinant by multilinearity. Terms with a repeated $q$ vanish by alternation. For each remaining index set $S=\{s_1<\cdots <s_r\}$, collect all permutations $q_j=s_{\sigma (j)}$. The factor from $A$ is $\mathrm{sgn}(\sigma )\det A_{:,S}$; the sum of the products $\mathrm{sgn}(\sigma ){\prod }_j{B}_{s_{\sigma (j)},j}$ is $\det B_{S,:}$. This proves the formula. Applying it to selected rows and columns proves that a product of compatible totally nonnegative matrices is totally nonnegative.