# Determinant of the matrix whose elements are given by a specific formula.

Calculate the determinant of $A_{n\times n}=((a_{ij})_{1\le i\le n})_{1\le j\le n}$ where

$$a_{ij}=\frac{1}{i+j-1}$$

-
Thanks @DominicMichaelis for pointing out the typo. – Abhra Abir Kundu Mar 1 '13 at 13:45
@AbhraAbirKundu: of course you mean for square matrices as DM notes. Hint, try a 2x2, 3x3 and look for a pattern to generalize. – Amzoti Mar 1 '13 at 13:48
Did you try anything? My first idea would be calculating for small $n$ and try to find a system – Dominic Michaelis Mar 1 '13 at 13:49
Yes i did try for small $n$ like $n=2,3,4$ but couldn't find any pattern in those determinants. – Abhra Abir Kundu Mar 1 '13 at 13:52

## 1 Answer

This matrix is known as Hilbert Matrix, see here: http://en.wikipedia.org/wiki/Hilbert_matrix

-
Thanks @pritam for the resource. – Abhra Abir Kundu Mar 1 '13 at 13:56
@Abhra Abir Kundu: You are welcome, once you know the formula you can try to prove it by induction. – pritam Mar 1 '13 at 14:06