Solution:

By place value, $\underline{a}\,\underline{b}$ in base seven is $7a + b,$ and $\underline{b}\,\underline{a}$ in base nine is $9b + a.$ Set them equal: $7a + b = 9b + a,$ so $6a = 8b,$ that is $3a = 4b.$ The digits have to fit, with $a \le 6$ and $b \le 8,$ and the only pair that works is $a = 4, b = 3.$ So $a + b = 7.$ Therefore, the answer is A.