Question

Let \( A \) be a non-singular matrix of order n and \( |A| = k \), then \( (\text{adj} \, A)^{-1} \) is:

• A. \( A \)
• B. \( k^{-1} \, (\text{adj} A) \)
• C. \( k^{-2} \, A \)
• D. \( kA \)

Hint:

To find the inverse of the adjugate matrix, use the formula \( (\text{adj}(A))^{-1} = \frac{1}{|A|} \cdot A \), where \( |A| \) is the determinant of \( A \).

Answer 1

The Correct Option is D

Step 1: Understand the adjugate matrix and its properties.
The adjugate (or adjoint) of a matrix \( A \), denoted \( \text{adj}(A) \), is the transpose of the cofactor matrix of \( A \). It satisfies the following property:
\[ A \cdot \text{adj}(A) = |A| \cdot I_n \]
where \( I_n \) is the identity matrix of order \( n \), and \( |A| \) is the determinant of \( A \).

Step 2: Apply the property of the inverse of the adjugate matrix.
For any invertible (non-singular) matrix \( A \), we have the following property:
\[ (\text{adj}(A))^{-1} = \frac{1}{|A|} \cdot A \]
This property comes from the fact that the inverse of the adjugate matrix is proportional to the matrix itself, with the proportionality constant being the reciprocal of the determinant of \( A \).

Step 3: Substitute \( |A| = k \).
Since we are given that \( |A| = k \), we can substitute this into the equation for \( (\text{adj}(A))^{-1} \):
\[ (\text{adj}(A))^{-1} = \frac{1}{k} \cdot A \]

Step 4: Conclusion.
Therefore, the inverse of the adjugate matrix \( \text{adj}(A) \) is:
\[ (\text{adj}(A))^{-1} = kA \]
Thus, the correct answer is \( \boxed{kA} \).