Question

If 

then \( {Adj} (A) \) is equal to:

• A.
• B.
• C.
• D.

Hint:

For a \( 2 \times 2 \) matrix, the adjugate is found by swapping the diagonal elements and negating the off-diagonal elements.

Answer 1

The Correct Option is B

We are given a matrix \( A \) as follows: \[ A = \begin{bmatrix} \alpha & \beta \\ \gamma & \alpha \end{bmatrix} \] The adjugate (or adjoint) of a matrix is defined as the transpose of its cofactor matrix. To find \( \text{Adj}(A) \), we first need to calculate the cofactors of each element of the matrix \( A \). For a \( 2 \times 2 \) matrix: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] the adjugate \( \text{Adj}(A) \) is given by: \[ \text{Adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] Now, applying this formula to the matrix \( A \) given in the problem: \[ A = \begin{bmatrix} \alpha & \beta \\ \gamma & \alpha \end{bmatrix} \] the adjugate matrix is: \[ \text{Adj}(A) = \begin{bmatrix} \alpha & -\beta \\ -\gamma & \alpha \end{bmatrix} \] Thus, the correct answer is: \[ \text{Adj}(A) = \begin{bmatrix} \delta & -\beta \\ -\gamma & \alpha \end{bmatrix} \] where \( \delta = \alpha \).