Question

If \( A \), \( B \), and \( \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) \) are non-singular matrices of the same order, then the inverse of \[ A \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) B \] is equal to:

• A. \( AB^{-1} + A^{-1}B \)
• B. \( \text{adj}(B^{-1}) + \text{adj}(A^{-1}) \)
• C. \( \frac{1}{|A|B|} \left( \text{adj}(B) + \text{adj}(A) \right) \)
• D. \( AB^{-1} + BA^{-1} \)

Hint:

To simplify matrix expressions involving adjugates and inverses, always apply known properties of determinants and adjugates. Using the relationship between the adjugate of the inverse and the original matrix helps in reducing complex expressions.

Answer 1

The Correct Option is C

Step 1: We start by writing the given expression: \[ A \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) B \] Now, apply the property of adjugates for inverses: \[ A \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) B = B^{-1} \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) A^{-1} \] Step 2: By applying the properties of adjugates and their relation to inverses, we simplify the expression further: \[ = B^{-1} \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) A^{-1} \] This simplifies to: \[ B^{-1} \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) A^{-1} \] Step 3: Using the determinant and adjugate properties, we arrive at the final form: \[ \frac{1}{|A|B|} \left( \text{adj}(B) + \text{adj}(A) \right) \]