Find inverse matrix for Vandermonde matrix using Cramer's rule

Question

I want to prove that I can find the inverse matrix for a Vandermonde matrix using Cramer's rule.

Let $A$ be a Vandermonde matrix, and $b$ the first column of the identity matrix $I$.

When I solve $Ax=b$ with Cramer's rule, is it true that $x$ is the first column in $A^{-1}$?

So I get matrix $X$, and $AX = I$.

Unfortunately, I don't know whether it is right or not that $XA = I$?