If you’re doing a lot of simulations, optimizations or other muckity-muck, it’s easy to start writing down equations with matrices like $A^{-1}B$ inside of them. And, depending on the application, it might be a really good idea to precompute this matrix. But there’s that pesky inverse operation, and generally inverting a matrix is a no-no since it can easily land you in trouble when that matrix isn’t really very stable.

So, here’s a trick you can use in these situations to avoid explicitly inverting the matrix A.  It turns out that instead we can just solve the system $Ax=b$ a bunch of times in order to explicitly compute the product matrix $A^{-1}B$.

It’s pretty straightforward.  If we want to extract the $i$th column from $A^{-1}B$, we need to solve the equation $x = A^{-1}B e_i$, where $e_i$ is the indicator vector with $0$s everywhere except the $i$th entry, which is $1$.  If we just massage this equation a bit, we get $Ax = Be_i = b_i$ where $b_i$ is the $i$th column of $B$.  Sweet!  If we have a good way to pre-factor $A$ or otherwise prepare it for repeated solves, we can also keep this solution method pretty efficient.