How to compute the inverse of a rank-$1$ matrix

Question

I have a rank-1 matrix $R \in \mathcal{C}^{m \times m}$, how to compute another matrix X, such that $RX=I$, where $I$ is an identity matrix.

Answer 1

This is impossible in general: it would mean the associated linear map $\;r\colon\mathbf C^m \longrightarrow \mathbf C^m\; $ is surjective. As the rank of the matrix is $1$, ie. the image of $r$ has dimension $1$, this is possible only if $m=1$.

Another way of proving it is to observe that $$\operatorname{rank}(RX)\le \operatorname{rank}R \le 1.$$