Does the Cauchy-Binet theorem simplify for matrices AWB, where W is a square matrix? Suppose I have two matrices $A$ and $B$, where $A$ is $m\times n$ and $B$ is $n\times m$. The Cauchy-Binet theorem gives a way to calculate $\det(AB)$:
$$\det(AB) = \sum_S\det(A_S)\det(B_S),$$
where the sum is over all length-$m$ subsets of $\{1,2,...,n\}$, $A_S$ is the $m\times m$ matrix whose columns are the columns in $A$ with indices $S$, and $B_S$ is the $m\times m$ matrix whose rows are the rows in $B$ with indices $S$.
Now suppose I have a square $n\times n$ matrix $W$. Is there a comparatively simple way to calculate $\det(AWB)$? It would be extra convenient if there is an expression for this in terms of $\det W$, so it could be extended to $N$ square matrices, something like $\det(A\prod_k^NW_kB)$.
I expected to find a corollary to the Cauchy-Binet theorem for linear transformations but haven't come across anything. Maybe it's obvious and my linear algebra is just too rusty?
 A: A more general statement of Cauchy-Binet says that if $A$ is $m\times n$ and $B$ is $n\times p$, then
$$(AB)_{IJ}=\sum_K A_{IK}B_{KJ}$$
where $(AB)_{IJ}$ denotes the minor determinant of $AB$ with rows indexed from $I\subseteq\{1,\ldots,m\}$ and columns indexed from $J\subseteq\{1,\ldots,p\}$, with $I$ and $J$ of same size. In the sum, $K$ ranges over all subsets of $\{1,\ldots,n\}$ of the same size as $I$ and $J$. Note if $m=p$ and $I=J=\{1,\ldots,m\}$, then this reduces to the version you stated.
In the language of exterior algebra, this result is equivalent to functoriality of $k$-th exterior powers, whose entries are $k\times k$ minors:
$$\textstyle\bigwedge^k(AB)=\bigwedge^k A\circ\bigwedge^k B$$
This extends to more than two matrices, so if $m=p$ and $W$ is $n\times n$, then
$$\textstyle\bigwedge^m(AWB)=\bigwedge^m A\circ\bigwedge^m W\circ\bigwedge^m B$$
The left hand side here is a $1\times 1$ matrix consisting of the single entry $\det(AWB)$, while the right hand side is a product of:

*

*the $1\times\binom{n}{m}$ matrix of $m\times m$ minors of $A$

*the $\binom{n}{m}\times\binom{n}{m}$ matrix of $m\times m$ minors of $W$

*the $\binom{n}{m}\times 1$ matrix of $m\times m$ minors of $B$
Example: if
$$A=\begin{bmatrix}9&2&2\\1&8&0\end{bmatrix}\qquad W=\begin{bmatrix}1&4&2\\1&1&0\\1&3&1\end{bmatrix}\qquad B=\begin{bmatrix}6&4\\0&9\\5&5\end{bmatrix}$$
then
$$AWB=\begin{bmatrix}178&548\\64&154\end{bmatrix}$$
so
$$\det(AWB)=-7660$$
The compound matrices of $2\times 2$ minors are:
$$\textstyle\bigwedge^2 A=\begin{bmatrix}70&-2&-16\end{bmatrix}\qquad\bigwedge^2 W=\begin{bmatrix}-3&-2&-2\\-1&-1&-2\\2&1&1\end{bmatrix}\qquad\bigwedge^2 B=\begin{bmatrix}54\\10\\-45\end{bmatrix}$$
the product of which is
$$\begin{bmatrix}-7660\end{bmatrix}$$
