I would like to prove, for any integer $n>1$, the invertibility of the $n\times n$ matrix $A$ whose elements are given by $A_{ij}=|i-j|^3$, where $i$ and $j$ are the indices.
To be clearer, for instance if $n=5$ I'm referring to the matrix $$A=\begin{bmatrix}&0&1&8&27&64\\&1&0&1&8&27\\&8&1&0&1&8\\&27&8&1&0&1\\&64&27&8&1&0\end{bmatrix}\,.$$
Such matrices should indeed be invertible (or at least it seems so looking at the determinant with Mathematica) for any $n>1$ and I do believe that there's should be some easy way to show it. I had some look on invertibility for Toeplitz or Hankel matrices, but I couldn't find any helps there for now.
Actually I guess I've found a way to tackle the problem. Defining $q$ the matrix with elements $q_{ij}=|i-j|$, which is not hard to prove to be invertible, one has $$q^{-1}\,A\,q^{-1}=M\,,$$ where $M$ is a sparse matrix (nearly tridiagonal), which can be easily shown to be invertible.
Now I'm looking for a smart way to prove that $M$ has such a simple structure. It's not hard to do it by direct calculations but it's a bit tedious and I'm sure there must be some smarter way to do it.