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} = ? $
Well, we could say that if $A+B$ is invertible,
$$
(A^{-1} + B^{-1})^{-1} = B(A+B)^{-1}A
$$
so that
$$
A^{-1} + B^{-1} + C^{-1} = \\
(A^{-1} + B^{-1}) + C^{-1} = \\
(A^{-1} + B^{-1})^{-1}((A^{-1} + B^{-1})^{-1} + C)C^{-1} =\\
B(A+B)^{-1}A(B(A+B)^{-1}A + C)C^{-1}
$$
yes, we can generalise.

$$ A^{-1}+B^{-1}+C^{-1}= BB^{-1}CC^{-1}A^{-1}+AA^{-1}B^{-1}CC^{-1}+AA^{-1}BB^{-1}C^{-1}$$
$$ =A^{-1}B^{-1}C^{-1}(AB+BC+CA) $$
That is, sum of the n inverse matrices is equal to product of the n inverse matrices and sum of the original matrices taken (n-1) at a time.