On invertibility of a special matrix - Hilbert matrix

Question

I want to know how to prove that the below matrix is invertible

\begin{pmatrix} 1 & \frac { 1 }{ 2 } & ... & \frac { 1 }{ n } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 3 } & ... & \frac { 1 }{ n+1 } \\ ... & ... & ... & ... \\ \frac { 1 }{ n } & \frac { 1 }{ n+1 } & ... & \frac { 1 }{ 2n-1 } \end{pmatrix}

and that the inverse matrix has integer entries.

I believe this is a known matrix with a wiki article. Can't remember its name, though. This has also been asked before. — Git Gud, Sep 29 '13 at 15:59
Try this and this. If you happen to find your question, please add a comment here so we can mark it as a duplicate. I suggest you add the name 'Hilbert Matrix' to the title somewhere. — Git Gud, Sep 29 '13 at 16:04

score 1 · Accepted Answer · answered Sep 29 '13 at 16:05

1

Here is the the article on Wikipedia with this matrix.

answered Sep 29 '13 at 16:05

Sasha Patotski

7,161

The problem came in the first chapter of a linear algebra book which I am trying to work through. They didn't mention the name of the matrix. Thanks for the link! – subrat verma Sep 29 '13 at 16:10

On invertibility of a special matrix - Hilbert matrix

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