Inverse of the sum of matrices I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-1}$ is not known, but if it is necessary then it can be assumed that $B^{-1}$ is also known.

This I found accidentally.
Suppose given $A$, and $B$, where $A$ and $A+B$ are invertible. Now we want to know the expression of $(A+B)^{-1}$ without imposing the all inverse. Now we follow the intuition like this. Suppose that we can express $(A+B)^{-1} = A^{-1} + X$, next we will present simple straight forward method to compute $X$
\begin{equation}
(A+B)^{-1} = A^{-1} + X
\end{equation}
\begin{equation}
(A^{-1} + X) (A + B) = I 
\end{equation}
\begin{equation}
A^{-1} A + X A + A^{-1} B + X B = I 
\end{equation}
\begin{equation}
X(A + B)  = - A^{-1} B
\end{equation}
\begin{equation}
X  = - A^{-1} B ( A + B)^{-1}
\end{equation}
\begin{equation}
X  = - A^{-1} B (A^{-1} + X)
\end{equation}
\begin{equation}
(I + A^{-1}B) X  = - A^{-1} B A^{-1} 
\end{equation}
\begin{equation}
X  = - (I + A^{-1}B)^{-1} A^{-1} B A^{-1}
\end{equation}
This lemma is simplification of lemma presented by Ken Miller, 1981
