sum of three inverse matrices

Question

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} = ? $

Answer 1

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

Answer 2

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.