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I am working on inverse matrices and I have raised a problem about the inverse of the sum of two matrix. I have searched for quite a while but still could not find an answer. Does anyone know what is $(A+B)^{-1}$? Is it just $A^{-1} + B^{-1}$ or something else?

Rodrigo de Azevedo
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user35263
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  • If you multiply $(A+B)(A^{-1}+B^{-1})$, it's clear it cannot be true in general. – Andrew Apr 12 '23 at 21:21
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    Note that even for scalars, one does not have $(a + b)^{-1} = a^{-1} + b^{-1}$.However, if your question is whether the inverse distributes across sums, the answer is no. The answer to your question is that if $(A+B)^{-1}$ exists, it is the unique matrix $M$ such that $M (A + B) = (A+ B)M = I$. – Drew Brady Apr 12 '23 at 21:31

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Note that the sum can be non-invertible for instance $A=I$ and $B=-I$.

Assuming that's not the case there is The woodbury matrix identity

Crayzyfrog
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    Woodbury matrix identity would give $$(A+C)^{-1}=A^{-1}-A^{-1}(A^{-1}+C^{-1})^{-1}A^{-1}$$ Is it easy for calculations ? – Jean Marie Apr 12 '23 at 21:29