Question

If \(A\) is a non singular matrix, then \(\text{Adj}(A^{-1}) =\)

• A. \((\text{Adj} A)^{-1}\)
• B. \(\frac{1}{|A|} A^{-1}\)
• C. \(|A| A^{-1}\)
• D. \(|A| A\)

Hint:

The adjugate of an inverse can be challenging. Remember the key matrix identities and operations such as \(A \cdot \text{Adj}(A) = |A| I\) which are crucial in manipulating expressions involving adjugates and determinants.

Answer 1

The Correct Option is A

For a non singular matrix \(A\), the adjugate of the inverse of \(A\), denoted as \(\text{Adj}(A^{-1})\), can be derived from the relationship: \[ A \cdot \text{Adj}(A) = |A| I \] where \(I\) is the identity matrix and \(|A|\) is the determinant of \(A\). By applying the properties of the adjugate and inverse matrices, and using the formula for the inverse of a product, we have: \[ \text{Adj}(A^{-1}) = \text{Adj}((\text{Adj}(A) \cdot |A|^{-1} I)^{-1}) = (\text{Adj}(A) \cdot |A|^{-1})^{-1} \] Since \(\text{Adj}(A) \cdot |A|^{-1}\) yields \(A^{-1}\), taking the inverse of this expression gives: \[ (\text{Adj}(A) \cdot |A|^{-1})^{-1} = (\text{Adj}(A))^{-1} \cdot |A| I = (\text{Adj}(A))^{-1} \] This leads us to conclude that \(\text{Adj}(A^{-1}) = (\text{Adj} A)^{-1}\), confirming the correct option.