Q:

Multiplicative Inverses: Abstract Algebra

I’m currently struggling to answer this question. Any help would be greatly appreciated! The question is as follows:
Let $n$ be a positive integer and let $a \in \mathbb{Z}_n$. Show that if $a$ has a multiplicative inverse $b$ (written $b = a^{−1}$), then $b$ is also invertible and $a = b^{−1}$ (i.e $(a^{−1})^{−1} = a$).

A:

If $a$ has a multiplicative inverse $b$, then by definition $a*b=e=b*a$,
so also $b*a=e=a*b$, so $b$ has an inverse and it's $a$.

