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 \( \delta = \alpha \).
