Inverse of an Inverse

 For any non-singular matrix $$M$$, we know that $$M^{-1} = \frac{1}{|M|}\operatorname{adj}(M)$$, which can be rewritten as:
$$\operatorname{adj}(M) = |M|M^{-1}$$
Substitute $$M = A^{-1}$$ and $$M = B^{-1}$$ into the property:
$$\operatorname{adj}(A^{-1}) = |A^{-1}|(A^{-1})^{-1} = \frac{1}{|A|}A$$
$$\operatorname{adj}(B^{-1}) = |B^{-1}|(B^{-1})^{-1} = \frac{1}{|B|}B$$

$$X = A \left( \frac{1}{|A|}A + \frac{1}{|B|}B \right)^{-1} B$$
 ($$(XYZ)^{-1} = Z^{-1}Y^{-1}X^{-1}$$):
$$X^{-1} = \left[ A \left( \frac{1}{|A|}A + \frac{1}{|B|}B \right)^{-1} B \right]^{-1}$$
$$X^{-1} = B^{-1} \left[ \left( \frac{1}{|A|}A + \frac{1}{|B|}B \right)^{-1} \right]^{-1} A^{-1}$$
$$X^{-1} = B^{-1} \left( \frac{1}{|A|}A + \frac{1}{|B|}B \right) A^{-1}$$
$$X^{-1} = \frac{1}{|A|} B^{-1} A A^{-1} + \frac{1}{|B|} B^{-1} B A^{-1}$$
Since $$A A^{-1} = I$$ and $$B^{-1} B = I$$:
$$X^{-1} = \frac{1}{|A|} B^{-1} + \frac{1}{|B|} A^{-1}$$
Replace $$B^{-1}$$ with $$\frac{\operatorname{adj}(B)}{|B|}$$ and $$A^{-1}$$ with $$\frac{\operatorname{adj}(A)}{|A|}$$:
$$X^{-1} = \frac{1}{|A|} \left( \frac{\operatorname{adj}(B)}{|B|} \right) + \frac{1}{|B|} \left( \frac{\operatorname{adj}(A)}{|A|} \right)$$
$$X^{-1} = \frac{\operatorname{adj}(B)}{|A||B|} + \frac{\operatorname{adj}(A)}{|A||B|}$$
$$X^{-1} = \frac{1}{|AB|} (\operatorname{adj}(B) + \operatorname{adj}(A))$$