Given that \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) and adj \(A = A\).

We know that for a 2x2 matrix, adj \(A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\).

Since adj \(A = A\), we have:

\(\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)

Comparing the elements, we get:

1. \(d = a\)

2. \(-b = b \Rightarrow b = 0\)

3. \(-c = c \Rightarrow c = 0\)

4. \(a = d\)

Now, we need to find the value of \(a + b + c + d\).

Substituting the values we found:

\(a + b + c + d = a + 0 + 0 + a = 2a\)