This is perhaps a straightforward question but I'm a little confused. An $n\times n$ Cauchy matrix $A$ is a matrix with entries $$a_{i,j}=\frac{1}{x_i-y_j}$$ for $1\le i,j\le n$, where $x_i$ and $y_j$ are elements of a field $\mathcal{F}$, and $(x_i)$ and $(y_j)$ are injective sequences (they do not contain repeated elements; elements are distinct).
The determinant of $A$ is given explicitly as $$\frac{\prod_{i=2}^n\prod_{j=1}^{i-1}(x_i-x_j)(y_j-y_i)}{\prod_{i=1}^n\prod_{j=1}^n(x_i-y_j)}.$$
My question is this: what happens if one replaces the denominator of $a_{i,j}$ with its absolute value, in other words, does the determinant of the matrix $A'$ where $$a'_{i,j}=\frac{1}{|x_i-y_j|}$$ have a similar explicit evaluation?