# Elementary proof that the Hilbert Matrix is invertible with integer entries [duplicate]

Possible Duplicate:
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?

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## marked as duplicate by Andrey Rekalo, Martin Brandenburg, Qiaochu Yuan, Pete L. Clark, Andy PutmanMar 31 '11 at 1:54

The Wikipedia article gives an explicit, and not terribly complicated, expression for the entries of the inverse. Can't you just multiply the two matrices together to get a proof? – Emil Jeřábek Mar 29 '11 at 18:15
This came up before: mathoverflow.net/questions/47561/… – Faisal Mar 29 '11 at 19:06

You can look at M.-D. Choi's paper

http://www.jstor.org/stable/pdfplus/2975779.pdf

(American Math Monthly, 1983, "Tricks or Treats with the Hilbert Matrix") for this, and much more.

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Freely avaliable online: vigo.ime.unicamp.br/HilbertMatrix.pdf – darij grinberg Mar 29 '11 at 19:20
The solution provided in the paper uses the concept of the determinant, but this question is provided as an exercise problem in Hoffman and Kunze's book way before the determinant is introduced. Before presenting the determinant solution, the above paper refers to the direct computation as "cumbersome". – fmardini Mar 29 '11 at 22:09
Suffering is good for you:) – Igor Rivin Mar 29 '11 at 22:14

Look at the paper below for what you are asking and much more! There is a lot known about the Hilbert matrix.

Tricks or Treats with the Hilbert Matrix by Man-Duen Choi published at The American Mathematical Monthly, Vol. 90, No. 5 (May, 1983), pp. 301-312

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Freely avaliable online: vigo.ime.unicamp.br/HilbertMatrix.pdf – darij grinberg Mar 29 '11 at 19:18