1. Key Concepts and Formulas

This problem requires a strong understanding of fundamental properties of non-singular matrices, particularly involving inverses and adjoints. We will use the following key properties:

• Inverse of a Product: For non-singular matrices $P, Q, R$ of the same order, $(PQR)^{-1} = R^{-1}Q^{-1}P^{-1}$.
• Relationship between Adjoint and Inverse: For a non-singular matrix $P$, $\operatorname{adj}(P) = |P|P^{-1}$. Consequently, $P^{-1} = \frac{1}{|P|}\operatorname{adj}(P)$.
• Adjoint of an Inverse: For a non-singular matrix $P$, $\operatorname{adj}(P^{-1}) = |P^{-1}|(P^{-1})^{-1} = \frac{1}{|P|}P$.
• Inverse of an Inverse: For a non-singular matrix $P$, $(P^{-1})^{-1} = P$.
• Determinant of a Product: For non-singular matrices $P, Q$, $|PQ| = |P||Q|$.

2. Step-by-Step Solution

Let the given expression be $X = A\left(\operatorname{adj}\left(A^{-1}\right)+\operatorname{adj}\left(B^{-1}\right)\right)^{-1} B$. We need to find $X^{-1}$.

Step 1: Simplify the adjoint terms using the property $\operatorname{adj}(P^{-1}) = \frac{1}{|P|}P$. The expression contains $\operatorname{adj}(A^{-1})$ and $\operatorname{adj}(B^{-1})$. We can simplify these: $\operatorname{adj}(A^{-1}) = \frac{1}{|A|}A$ $\operatorname{adj}(B^{-1}) = \frac{1}{|B|}B$ This simplification is crucial as it replaces the adjoint of an inverse with a simpler form involving the original matrix and its determinant.

Step 2: Substitute the simplified adjoint terms into the expression. Let $M = \operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})$. Substituting the simplified forms: $M = \frac{1}{|A|}A + \frac{1}{|B|}B$ Now, the expression for which we need to find the inverse becomes: $X = A M^{-1} B$

Step 3: Apply the inverse of a product rule to find $X^{-1}$. We need to find $X^{-1} = (A M^{-1} B)^{-1}$. Using the property $(PQR)^{-1} = R^{-1}Q^{-1}P^{-1}$: $X^{-1} = B^{-1} (M^{-1})^{-1} A^{-1}$ Using the property $(P^{-1})^{-1} = P$, we have $(M^{-1})^{-1} = M$. $X^{-1} = B^{-1} M A^{-1}$ This step breaks down the overall inverse into a product of simpler inverses, making it easier to substitute back the expression for $M$.

Step 4: Substitute the expression for $M$ back into $X^{-1}$ and simplify. $X^{-1} = B^{-1} \left( \frac{1}{|A|}A + \frac{1}{|B|}B \right) A^{-1}$ Now, distribute $B^{-1}$ from the left and $A^{-1}$ from the right: $X^{-1} = B^{-1} \left(\frac{1}{|A|}A\right) A^{-1} + B^{-1} \left(\frac{1}{|B|}B\right) A^{-1}$ Rearrange the scalar and matrix terms: $X^{-1} = \frac{1}{|A|} (B^{-1} A A^{-1}) + \frac{1}{|B|} (B^{-1} B A^{-1})$ Since $A A^{-1} = I$ (identity matrix) and $B^{-1} B = I$: $X^{-1} = \frac{1}{|A|} B^{-1} I + \frac{1}{|B|} I A^{-1}$ $X^{-1} = \frac{1}{|A|} B^{-1} + \frac{1}{|B|} A^{-1}$ This is the simplified form of the inverse.

Step 5: Compare the result with the given options by simplifying the options themselves. Let's check option (D): $\frac{1}{|A B|}(\operatorname{adj}(B)+\operatorname{adj}(A))$ Using the properties $|AB| = |A||B|$ and $\operatorname{adj}(P) = |P|P^{-1}$: $\frac{1}{|A B|}(\operatorname{adj}(B)+\operatorname{adj}(A)) = \frac{1}{|A||B|} (|B|B^{-1} + |A|A^{-1})$ Distribute $\frac{1}{|A||B|}$ into the parenthesis: $= \frac{|B|B^{-1}}{|A||B|} + \frac{|A|A^{-1}}{|A||B|}$ $= \frac{B^{-1}}{|A|} + \frac{A^{-1}}{|B|}$ This matches our derived expression for $X^{-1}$.

3. Common Mistakes & Tips

• Misinterpreting Adjoint Properties: A common mistake is to confuse $\operatorname{adj}(P^{-1})$ with $(\operatorname{adj} P)^{-1}$. Remember, $\operatorname{adj}(P^{-1}) = \frac{1}{|P|}P$, while $(\operatorname{adj} P)^{-1} = \left(|P|P^{-1}\right)^{-1} = \frac{1}{|P|} (P^{-1})^{-1} = \frac{1}{|P|}P$. In this specific case, they are the same. However, a common mistake is to forget the determinant factor. The key property is $\operatorname{adj}(P) = |P|P^{-1}$.
• Order of Operations for Inverse of a Product: Always remember that $(PQR)^{-1} = R^{-1}Q^{-1}P^{-1}$. The order of matrices is reversed when taking the inverse of a product.
• Careless Distribution: When distributing matrix terms, ensure to maintain the correct order of multiplication (e.g., $B^{-1} \left(\frac{1}{|A|}A\right) A^{-1}$ is not the same as $\left(\frac{1}{|A|}A\right) B^{-1} A^{-1}$).

4. Summary

To find the inverse of the complex matrix expression, we first simplified the adjoint terms within the parenthesis using the property $\operatorname{adj}(P^{-1}) = \frac{1}{|P|}P$. This transformed the expression into a product of three matrices. We then applied the inverse of a product rule, $(PQR)^{-1} = R^{-1}Q^{-1}P^{-1}$, and the inverse of an inverse property, $(P^{-1})^{-1} = P$. Finally, we substituted the simplified terms back and performed matrix multiplication, utilizing $PP^{-1} = I$, to arrive at the simplified inverse. Comparing this result with the given options by expanding them revealed a match with option (D).

5. Final Answer

The inverse of $A\left(\operatorname{adj}\left(A^{-1}\right)+\operatorname{adj}\left(B^{-1}\right)\right)^{-1} B$ is $\frac{1}{|A|} B^{-1} + \frac{1}{|B|} A^{-1}$, which can also be written as $\frac{1}{|A B|}(\operatorname{adj}(B)+\operatorname{adj}(A))$.

The final answer is \boxed{\frac{1}{|A B|}(\operatorname{adj}(B)+\operatorname{adj}(A))}, which corresponds to option (D).