Step 1 :Given the surface area of sphere B is 7 km^2 and the scale factor of dilation is \(\frac{1}{3}\).
Step 2 :Set up a proportion between the surface areas of sphere A and sphere B: \(\frac{S_A}{S_B} = \left(\frac{1}{3}\right)^2\).
Step 3 :Plug in the given value of $S_B$: \(\frac{S_A}{7} = \left(\frac{1}{3}\right)^2\).
Step 4 :Solve for $S_A$: \(S_A = 7 \cdot \left(\frac{1}{3}\right)^2\).
Step 5 :Calculate the surface area of sphere A: \(S_A \approx 0.78 \mathrm{km}^2\).
Step 6 :\(\boxed{S_A \approx 0.78 \mathrm{km}^2}\)