Assuming that there exists a transitive model of ZF(C). Show that there exists a minimal transitive model of ZF(C).
I am not quite sure how to approach this problem, how do I use the fact that $\kappa$ is weakly inaccessible to construct a minimal transitive model of ZF(C)?
The fact is that you just need a transitive model. Here are some lemmas that will help you figure this out:
If $M$ is a transitive set model, then $L^M$ is some $L_\alpha$ for some ordinal $\alpha$ which is also $M\cap\rm Ord$. And $L^M$ is the smallest model of height $\alpha$.
If there is at least one $\alpha$ such that $L_\alpha$ is a model of $\sf ZFC$, then there is a least such $\alpha$.
Indeed, the weakly inaccessible cardinal is sufficient, but it is far stronger than what is absolutely necessary.