---
tag: 0P84B554AS
kind: statement
role: theorem
name: Vandermonde determinant
msc: [15A15, 15B05]
uses: [0C3CFHDN6R]
sources:
  - url: https://en.wikipedia.org/wiki/Vandermonde_matrix
    locator: Determinant
---

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $x_1, \dots, x_n \in K$.
Let $V$ be the $n \times n$ *Vandermonde matrix* with entries $V_{ij} = x_i^{\,j-1}$.
Then
$$\det V = \prod_{1 \le i < j \le n} (x_j - x_i),$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Over a field, $\det V \ne 0$ if and only if the $x_i$ are pairwise distinct.
Then, for all $y_1, \dots, y_n$, there is exactly one polynomial $p$ of degree less than $n$ with $p(x_i) = y_i$ for all $i$.
For $n = 2$, the matrix is $\begin{pmatrix} 1 & x_1 \\ 1 & x_2 \end{pmatrix}$ with determinant $x_2 - x_1$.

## Explanations

### L1 human

A Vandermonde matrix lists the powers $1, x, x^2, \dots$ of some numbers, one row for each number.
Its determinant is the product of all differences of the numbers.
So it is nonzero exactly when the numbers are distinct.