  We consider fluctuations of the largest eigenvalues of the random matrix
model $A+UBU^{*}$ where $A$ and $B$ are $N \times N$ deterministic Hermitian
(or symmetric) matrices and $U$ is a Haar-distributed unitary (or orthogonal)
matrix. We prove that the largest eigenvalue weakly converges to the
Tracy-Widom distribution, under mild assumptions on $A$ and $B$ to guarantee
that the density of states of the model decays as square root around the upper
edge. Our proof is based on the comparison of the Green function along the
Dyson Brownian motion starting from the matrix $A + UBU^{*}$ and ending at time
$N^{-1/3+\chi}$. As a byproduct of our proof, we also prove an optimal local
law for the Dyson Brownian motion up to the constant time scale.
