$a$ is unit in $\mathbb{Z}$ if and only if $-a$ is unit

Question

By definition I have that if $a\in\mathbb{Z}$ has a multiplicative inverse then $a$ is unit.

I need to prove that $a$ is unit if and only if $-a$ is unit. Assuming that $a$ is unity then there exists $b\in\mathbb{Z}$ such that $ab=1$. To prove the result, it suffices to prove that if $ab=1$ then $a=b=1$ or $a=b=-1$ ?

I have tried this and also that $1*1=(-1)*(-1)$, but it is not clear to me, taking into account that it is an if and only if, I don't know how to deal with both cases.

Answer 1

Assume $a$ is a unit. We need to show that $-a$ is a unit. Since $a$ is a unit we have $ab=1$ for some $b\in\mathbb{Z}$. Now observe $(-a)(-b)=ab=1$, so that $-a$ is a unit. A symmetric argument shows that if $-a$ is a unit, then $a$ is a unit.