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: Understanding the Question:
We need to find an expression for the inverse of the adjoint of a non-singular matrix \( \text{A} \).
Step 2: Key Formula or Approach:
Use the property \( \text{A}^{-1} = \frac{1}{|A|} \text{adj A} \) and its manipulations.
Step 3: Detailed Explanation:
1. We know that \( A \cdot \text{adj A} = |A| \text{I} \).
2. Since \( A \) is non-singular, \( \text{adj A} \) is also invertible. Multiply by \( (\text{adj A})^{-1} \) from the right:
\( A \cdot (\text{adj A}) \cdot (\text{adj A})^{-1} = |A| \text{I} \cdot (\text{adj A})^{-1} \).
3. \( A = |A| (\text{adj A})^{-1} \).
4. Given \( |A| = k \), we have:
\( A = k (\text{adj A})^{-1} \).
5. Rearranging for \( (\text{adj A})^{-1} \):
\( (\text{adj A})^{-1} = \frac{A}{k} \).
Step 4: Final Answer:
The value of \( (\text{adj A})^{-1} \) is \( \frac{A}{k} \).