Efficient way of computing $A^{-1}B$ knowing $A$ and $BA$

Question

Let $A$ and $B$ be two invertible matrices $n \times n$ matrices, for some positive integer $n$. Knowing the matrices $A$ and $BA$ (that is, specifically knowing their entries), how can we efficiently compute $A^{-1}B$?

Here, by efficiently, I mean using an efficient algorithm (so computing the inverse of $A$ does not seems like a good idea).

I thought about finding a relation between $A,BA$ and $A^{-1}B$, but I have not succeeded (a relation such at $A^{-1}B = CA + DBA$ for some matrices $C$ and $D$ that we can easily find out).

Another idea is to use the $QR-$decomposition of the matrices $A$ and $B$, because we can easily find the inverse of an orthogonal matrix and the inverse of an upper triangular matrix, but we don't specifically know the matrix $B$, and the complexity is $O(n^3)$, which doesn't seem like the most efficient way.

Answer 1

You could do:

1. Row reduce $[A^t|(BA)^t]=[A^t|A^tB^t]$ to $[I|B^t]$, giving you $B$.
2. Row reduce $[A|B]$ to $[I|A^{-1}B]$

Does that count as efficient?