Among the given species. \( \text{S}_2\text{O}_3^{2-} \) (thiosulfate ion) has a pyramidal geometry around the central sulfur atom due to the presence of a lone pair on sulfur, resulting in a trigonal pyramidal shape.\( \text{SO}_4^{2-} \) (sulfate ion) has a tetrahedral geometry due to its symmetrical distribution of oxygen atoms around the central sulfur atom.\( \text{SO}_3^{2-} \) (sulfite ion) also adopts a trigonal pyramidal structure around sulfur, but in this context, \( \text{S}_2\text{O}_3^{2-} \) is more relevant to pyramidal geometry.\( \text{S}_2\text{O}_7^{2-} \) (disulfate ion) has a tetrahedral arrangement with respect to its central atoms, making it non-pyramidal.
Therefore, only one species, \( \text{S}_2\text{O}_3^{2-} \), exhibits pyramidal geometry.
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: