Step 1: Understand the wave.
The wave is $y = A\sin(kx - \omega t)$. Here $y$ is how far a particle moves, $A$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency.
Step 2: Know particle velocity.
The particle velocity is how fast a point on the string moves up and down. We get it by differentiating $y$ with respect to time $t$, keeping $x$ fixed.
Step 3: Differentiate the wave.
\[ v_y = \frac{\partial y}{\partial t} = A\cos(kx-\omega t)\cdot(-\omega) \] So \[ v_y = -A\omega\cos(kx-\omega t) \]
Step 4: Find the maximum.
The cosine term swings between $-1$ and $+1$. The speed is largest when the cosine reaches its peak value of $1$.
Step 5: Write the maximum value.
Putting the cosine equal to one, \[ v_{y,max} = A\omega \]
Step 6: State the answer.
So the largest particle velocity is the amplitude times the angular frequency. \[ \boxed{v_{y,max} = A\omega} \]