Question

For a matrix $ A $ of order $ 3 \times 3 $, which of the following is true?

• A. \( \text{adj}(A) = A^2 \)
• B. \( \text{adj}(A) \neq A^2 \)
• C. \( \text{adj}(A) = A^T \)
• D. \( \text{adj}(A) = A^{-1} \)

Hint:

The adjugate of a matrix \( A \) is related to its inverse when the determinant of \( A \) is non-zero. Specifically, \( A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A) \).

Answer 1

The Correct Option is D

For any square matrix \( A \), the adjugate (or adjoint) of the matrix \( A \), denoted \( \text{adj}(A) \), has the following property: \[ A \cdot \text{adj}(A) = |A| \cdot I \] Where:
- \( A \) is the matrix
- \( \text{adj}(A) \) is the adjugate matrix of \( A \)
- \( |A| \) is the determinant of \( A \)
- \( I \) is the identity matrix of the same order as \( A \)
For a matrix \( A \) of order \( 3 \times 3 \), when the determinant of \( A \) is non-zero, we can express the inverse of \( A \) as: \[ A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A) \] Therefore, for a matrix \( A \), the adjugate of \( A \) is related to the inverse of \( A \) by the equation: \[ \text{adj}(A) = A^{-1} \] Thus, the correct answer is \( \text{adj}(A) = A^{-1} \).