Let the initial rate of the reaction be $r_1 = k[A]^2[B]$. When the initial concentration of A is tripled, the new concentration becomes $[A]' = 3[A]$. The concentration of B is kept constant, so $[B]' = [B]$. The new rate is $r_2 = k[A]'^2[B]' = k(3[A])^2[B] = 9k[A]^2[B] = 9r_1$. Therefore, the initial rate increases by a factor of nine.