Column count from solvability of two linear systems

A matrix $A$ has $3$ rows and $n\ge2$ columns. Let $e_1,e_2,e_3$ be the standard basis of $\mathbb{R}^3$. The system $Ax=e_1$ is inconsistent, while $Ax=e_2$ has exactly one solution. Determine $n$.

$n=2$. A unique solution to $Ax=e_2$ forces the null space to be trivial, so $\operatorname{rank}A=n$. Inconsistency of $Ax=e_1$ means the column space is not all of $\mathbb{R}^3$, so $\operatorname{rank}A\le2$. Hence $2\le n\le2$.