Inverse of Vandermonde Matrix/Corollary

Define for variables $\set {y_1,\ldots,y_k}$ elementary symmetric functions:

Let $\set {x_1,\ldots,x_n}$ be a set of distinct values.

Let $W_n$ and $V_n$ be Vandermonde matrices of order $n$:



W_n = \begin{bmatrix} 1        &  x_1      & \cdots & x_1^{n-1} \\ 1        & x_2       & \cdots & x_2^{n-1} \\ \vdots     & \vdots    & \ddots & \vdots    \\ 1        & x_1^{n-1} & \cdots & x_n^{n-1} \\ \end{bmatrix},\quad V_n = \begin{bmatrix} x_1  & x_2    & \cdots & x_n    \\ x_1^2 & x_2^2 & \cdots & x_n^2  \\ \vdots & \vdots & \ddots & \vdots \\ x_1^n & x_2^n & \cdots & x_n^n  \\ \end{bmatrix} $

Let matrix inverses be written as $W_n^{-1} = \begin{bmatrix} b_{ij} \end{bmatrix}$ and $V_n^{-1} = \begin{bmatrix} c_{ij} \end{bmatrix}$. Then: