How to show that given language is unambiguous
To show a grammar is unambiguous you have to argue that for each string in the language there is only one derivation tree.

In this particular case you can observe that $A$ only generates $0$'s, so the $1$ generated by the start symbol $S$ must be the first $1$ in the string. 

Any grammar can be made ambiguous by adding chain productions like $S\to S$.