\[\det\left[\frac{1}{a_{j}-b_{k}}\right]=(-1)^{n(n-1)/2}\*\prod_{1\leq j<k\leq n%
}(a_{k}-a_{j})(b_{k}-b_{j})\Bigg/\prod^{n}_{j,k=1}(a_{j}-b_{k}).\]