Tyrosine is an amino acid with the following structure:
The structure contains:
A benzene ring (6 carbon atoms),
A carbon atom in the side chain connected to NH$_2$ and COOH,
Two additional carbon atoms in the COOH and CH$_2$OH groups.
Total carbon atoms = 9.
Let A be a 3 × 3 matrix such that \(\text{det}(A) = 5\). If \(\text{det}(3 \, \text{adj}(2A)) = 2^{\alpha \cdot 3^{\beta} \cdot 5^{\gamma}}\), then \( (\alpha + \beta + \gamma) \) is equal to: