Let $x_1, \dots, x_k \in \mathbb{R}^n$ be distinct points and let $A$ be the matrix defined by $A_{ij} = d(x_i, x_j)$, where $d$ is the Euclidean distance. Is $A$ always nonsingular?
I have a feeling this should be well known (or, at least a reference should exists), on the other hand, this fact fails for general metrics (take e.g. path metric on the cycle $C_4$)
edit: changed number of points from $n$ to general $k$