If I have invertible matrices $A, B, C$, whats the inverse of $(A + BC)$

Question

If matrices $A$, $B$ and $C$ are $n \times n$ invertible matrices, and assuming $A+BC$ is invertible, how do I find $$(A + BC)^{-1}.$$

https://math.stackexchange.com/questions/17776/inverse-of-the-sum-of-matrices related — b00n heT, Mar 10 '20 at 15:28
In general there is no magic formula that relates it to $A^{-1},B^{-1},C^{-1}$ without having any infinite sum or an inverse of a sum of matrices. So a good answer to this really depends on what exactly you want to do. — Ian, Mar 10 '20 at 15:36

score 1 · Answer 1 · answered Mar 10 '20 at 15:33

1

One possible answer is $$(A+BIC)^{-1}= A^{-1} - A^{-1}B(I +CA^{-1}B)^{-1}CA^{-1}.$$

If you write it as $(A+BCI)^{-1}$, you can get a different but equal form which may be more useful for some purposes.

answered Mar 10 '20 at 15:33

Is there any reason to expect that inverting $I+CA^{-1}B$ is any easier than inverting $A+BC$? – TonyK Mar 10 '20 at 15:37
Not in general. But in some cases, particularly in data science applications, this approach is useful. Assuming you have $A^{-1}$ precomputed, the original purpose of this formula is to update the inverse without calculating it from scratch. See https://en.wikipedia.org/wiki/Woodbury_matrix_identity#Applications – Abdullah Ali Sivas Mar 10 '20 at 15:43

score 0 · Answer 2 · answered Mar 10 '20 at 15:42

0

Nothing much better than $$(A+BC)^{-1}=A^{-1}(I+A^{-1}BC)^{-1}=A^{-1}\sum_{k=0}^\infty(-A^{-1}BC)^k,$$ if that sum converges.

answered Mar 10 '20 at 15:42

2 Answers2