The equilibrium reaction is:
\[ \text{Cr}_2\text{O}_7^{2-} + [\text{H}^+] + \text{OH}^- \rightleftharpoons 2\text{CrO}_4^{2-}. \]
In a basic medium (OH$^-$), the reaction shifts to the right to produce more CrO$_4^{2-}$, as OH$^-$ consumes H$^+$, reducing its concentration. Conversely, in an acidic medium (H$^+$), the equilibrium shifts to the left, favoring Cr$_2$O$_7^{2-}$.
Thus, the equilibrium shifts to the right in a basic medium.
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: