Given: \( A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)

We know that \( A \cdot (\text{adj } A) = |A| \cdot I \), where \( I \) is the identity matrix.

So, \( |A| \cdot I = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} = 3 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = 3I \)

Therefore, \( |A| = 3 \)

Now, we need to find \( |\text{adj } A| \). We know that \( |\text{adj } A| = |A|^{n-1} \), where \( n \) is the order of the matrix \( A \). In this case, \( n = 3 \).

So, \( |\text{adj } A| = |A|^{3-1} = |A|^2 = 3^2 = 9 \)

Finally, we need to find \( |A| + |\text{adj } A| = 3 + 9 = 12 \)