Hilbert Matrix

A Matrix ${\hbox{\sf H}}$ with elements

$$H_{ij} \equiv (i+j-1)^{-1}$$

for $i, j = 1$, 2, ..., $n$. Although the Matrix Inverse is given analytically by

$$(H^{-1})_{ij} = {(-1)^{i+j}\over i+j-1} {(n+i-1)!(n+j-1)!\over [(i-1)!(j-1)!]^2(n-i)!(n-j)!},$$

Hilbert matrices are difficult to invert numerically. The Determinants for the first few values of ${\hbox{\sf H}}_n$ are given in the following table.

$n$	det( ${\hbox{\sf H}}$)
1	1
2	$8.33333\times 10^{-2}$
3	$4.62963\times 10^{-4}$
4	$1.65344\times 10^{-7}$
5	$3.74930\times 10^{-12}$
6	$5.36730\times 10^{-18}$