Inverse of $(A + B)$ and $(A + BCD)$? Consider $A$ as an arbitrary matrix and $B$ as a symmetric matrix. Since $B$ is symmetric, therefore, it can be written as a $\Gamma \Delta \Gamma'$, where $\Delta$ is a diagonal matrix with eigen-values on the the main diagonal and $\Gamma$ is a matrix of corresponding eigenvectors.
Is there any formula for $(A + B)^{-1}$? 
Now, consider $A$, $B$, $C$, and $D$ as arbitrary matrices. Is there any formula for:
$(A + BCD)^{-1}$
Application: I'm fitting a model that involves matrix inversion. My matrix can be decomposed into two parts. At each step, only one part of the matrix gets updated, if I can find a formula for the inverse of the matrices above, then I don't need to invert the whole matrix and instead, I can only invert the part that has been updated.
Thanks for your help,

Have you considered the woodbury matrix inversion AKA Matrix Inversion Lemma?
https://en.wikipedia.org/wiki/Woodbury_matrix_identity
