jmc

For $n$ to have an inverse $(\mathrm{mod}1050)$, it is necessary for $n$ to be relatively prime to $1050$. Conversely, if $n$ is relatively prime to $1050$, then $n$ has an inverse $(\mathrm{mod}1050)$.

The prime factors of $1050$ include $2$, $3$, $5$, and $7$, so any multiple of any of these primes does not have an inverse $(\mathrm{mod}1050)$. This rules out all the integers from $2$ to $10$. However, $11$ is relatively prime to $1050$, so $\boxed{11}$ is the smallest integer greater than $1$ that has an inverse $(\mathrm{mod}1050)$.