Assume real rectangular matrices $A$ and $B$, where $A$ is $m \times n$, $B$ is $n \times m$, $m<n$, and the $m \times m$ product $AB$ is invertible. What are some possible strategies for simplifying the expression for the inverse $(AB)^{-1}$?
For example, the Cauchy-Binet formula provides an expression for $\det(AB)$ as a sum of products of simpler determinants. Is there a related formula for the inverse?
Are there more options available if we further assume that $AB=B^T B$? Or additionally that $AB=B^T B$ is a Gramian matrix?
By Cramer's Rule, $(AB)^{-1}_{ij} = (-1)^{i+j}\det(M_{ji})/\det(AB)$ where $M_{ji}$ is the matrix obtained by removing the $j$'th row and $i$'th column of $AB$. Thus $M_{ji} = A_j B_i$ where $A_j$ is obtained by removing the $j$'th row from $A$ and $B_i$ is obtained by removing the $i$'th column from $B$. You can then apply Cauchy-Binet to $\det(A_j B_i)$.