On page 90 of How to Think about Abstract Algebra, Lara Alcock says,

An element has an inverse if and only if every element appears in its row [in the multiplication table of a binary operation].

Let’s prove this. ($\Rightarrow$) Suppose $gg^{-1}=1$ and $a$ is an arbitrary element. We want to prove that $\exists h$ such that $gh=a$. Just pick $h=g^{-1}a$.

($\Leftarrow$) If $g$ is not invertible, then $1$ does not appear in its row.