Step 1: Write the hydrolysis
Na\(_2\)CO\(_3\) is a salt of a strong base (NaOH) and a weak acid (H\(_2\)CO\(_3\)).
The carbonate ion reacts with water:
\[ \text{CO}_3^{2-}+\text{H}_2\text{O}\rightleftharpoons\text{HCO}_3^-+\text{OH}^- \]
Step 2: Read off the result
\([\text{OH}^-]\) rises, so the solution is basic, pH \(>7\).
So (A) is right, (B) is wrong because pH changes, (C) is wrong because \([\text{H}^+]\) falls, and (D) is wrong because a basic solution turns red litmus blue, not blue litmus red.
Final Answer:
Dissolving Na\(_2\)CO\(_3\) raises \([\text{OH}^-]\), option (A).
\[ \boxed{\text{(A)}} \]