For matrices \(A=[a_{ij}]\), \(a_{ij}=[x_ i-x_ j]^{\alpha}\), where \(x_ 1,\dots,x_ n\) are pairwise distinct real numbers and \(\alpha >0\) not an even integer less than \(n-1\) it is shown that \((-1)^ V \det A>0,\) where \(V=0\) for n odd, \(V=n/2\) for n even and \(\alpha\geq n\), \(V=1+[\alpha /2]\) otherwise.
For \(\alpha\) an even integer less than n-1, det A\(=0\). This is proved by analyzing the spectrum of A and showing that the spectrum is independent of the choice of \(x_ i.\)
Oscillation properties for the eigenvectors of such matrices are also derived. Furthermore the case that \(x_ i\in {\mathbb{R}}^ 2\) and \(2<\alpha <4\) is studied and sufficient conditions for nonsingularity are obtained.