Deriving Inverse of Hilbert Matrix
The inverse of the Hilbert Matrix is made up entirely of integer entries, but I can't seem to find an elementary proof for that though, any hints?
You can look at M.-D. Choi's paper
(American Math Monthly, 1983, "Tricks or Treats with the Hilbert Matrix") for this, and much more.
Look at the paper below for what you are asking and much more! There is a lot known about the Hilbert matrix.