Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$

Question

Let $p$ be a prime and a an integer. Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$.

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Answer 1

Hint: $[a]^{-1} = [a] \iff 1 = [a]^2 \iff [a]^2 - 1 = 0 \iff ([a] - 1)([a] + 1) = 0$.