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

Given that matrices \( A \), \( B \), and \( \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) \) are non-singular, we aim to compute the inverse of the expression:

\(A \left( \text{adj}(A^{-1}) + \text{adj}(B^{-1}) \right) B\)

The relationship between the adjugate of the inverse of a matrix \( M \) and \( M \) itself is given by:

\(\text{adj}(M^{-1}) = |M| M\)

where \( |M| \) denotes the determinant of \( M \).

Applying this formula to \( A^{-1} \) and \( B^{-1} \):

\(\text{adj}(A^{-1}) = |A^{-1}| A\)

\(\text{adj}(B^{-1}) = |B^{-1}| B\)

Leveraging the property that \( |M^{-1}| = \frac{1}{|M|} \), we substitute:

\(\text{adj}(A^{-1}) = \frac{1}{|A|} A\)

\(\text{adj}(B^{-1}) = \frac{1}{|B|} B\)

Substituting these into the original expression \( A (\text{adj}(A^{-1}) + \text{adj}(B^{-1})) B \) yields:

\(A \left( \frac{1}{|A|} A + \frac{1}{|B|} B \right) B\)

This simplifies to:

\(= A \left( \frac{A}{|A|} + \frac{B}{|B|} \right) B\)

Further simplification leads to:

\(= \frac{1}{|A||B|} A (A + B) B\)

To find the inverse of this expression, we use the property \( (XY)^{-1} = Y^{-1} X^{-1} \).

Therefore, the inverse is:

\(\frac{1}{|A| B|} \left( \text{adj}(B) + \text{adj}(A) \right)\)

The correct option is:

\(\frac{1}{|A|B|} \left( \text{adj}(B) + \text{adj}(A) \right)\)