Consider $n$ distinct points in the plane $z_1, \ldots, z_n$. Form the matrix $D$ containing their squared distances as entries:
$$ D_{ij} \ = \ |z_i - z_j|^2 \, . $$
Obviously, this matrix is symmetric and non-negative and only the entries on the main diagonal are zero.
I am looking for necessary and sufficient conditions in terms of the $z_i$ for the matrix $D$ to be invertible and appropriate references.
The rule of sarrus implies $D$ to be invertible, if $n \ \leq \ 3$, but if 4 points form the corners of a rectangle, the corresponding $D$ is never invertible, while if three of the four points form an equilateral triangel, invertibility depends on the position of the fourth point. Therefore I fear the problem to be rather hopeless in general, but if any information is known, I'd like to have it.
I am interested in this type of information, because I know the image of the row vectors of this matrix when mapped by the so-called augmented Gauss-Lucas-matrix. If $D$ is invertible, I therefore know the corresponding augmented Gauss-Lucas matrix in principle.