<s>[INST] sum of three inverse matrices The following Searle identity computes the sum of two inverses:
$A^{-1}+B^{-1} = A^{-1}(A+B)B^{-1}$.
Is there any generalisation of this for the sum of three inverses?
$A^{-1}+B^{-1}+C^{-1} = ? $
 [/INST] Well, we could say that if $A+B$ is invertible,
$$
(A^{-1} + B^{-1})^{-1} = B(A+B)^{-1}A
$$
so that
$$
A^{-1} + B^{-1} + C^{-1} = \\
(A^{-1} + B^{-1}) + C^{-1} = \\
(A^{-1} + B^{-1})^{-1}((A^{-1} + B^{-1})^{-1} + C)C^{-1} =\\
B(A+B)^{-1}A(B(A+B)^{-1}A + C)C^{-1}
$$
</s>