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

The adjugate of a square matrix \( A \), denoted \( \text{adj}(A) \), satisfies the property: \[A \cdot \text{adj}(A) = |A| \cdot I\] Here, \( A \) is the matrix, \( \text{adj}(A) \) is its adjugate, \( |A| \) is its determinant, and \( I \) is the identity matrix of the same order. For a \( 3 \times 3 \) matrix \( A \) with a non-zero determinant, its inverse \( A^{-1} \) can be expressed as: \[A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)\] This implies that for any matrix \( A \), the adjugate is related to its inverse by the equation: \[\text{adj}(A) = A^{-1}\] Consequently, the correct relation is \( \text{adj}(A) = A^{-1} \).