If A,B are non-singular matrices of the same order, then the inverse of A(adj(A^-1)+adj(B^…

Question

If $A,B$ are non-singular matrices of the same order, then the inverse of $A\big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\big)^{-1}B$ is equal to:

Accepted Answer

Let $M=A\big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\big)^{-1}B$. Then
$$M^{-1}=B^{-1}\big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\big)A^{-1}.$$

Using $\operatorname{adj}(A^{-1})A^{-1}$-type identities and $A\cdot\operatorname{adj}(A)=|A|I$, one has $\operatorname{adj}(A^{-1})=\dfrac{\operatorname{adj}A}{|A|}$ and similarly for $B$. Substituting and simplifying:
$$M^{-1}=\frac{\operatorname{adj}B}{|B||A|}+\frac{\operatorname{adj}A}{|A||B|}=\frac{\operatorname{adj}A+\operatorname{adj}B}{|A||B|}.$$

So $M^{-1}=\dfrac{1}{|AB|}\big(\operatorname{adj}A+\operatorname{adj}B\big)$ --- Option D.

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