Rank of matrices with entries $1/(i+j)$ What is the rank of the $n$ by $n$ matrix $A$ whose coefficients are given by $a_{i,j}=1/(i+j)$?
Since this matrix $A$ is a Cauchy matrix for $x_i=i$ and $y_j=-j$, i.e., with
$a_{ij}=\frac{1}{x_i-y_j}=\frac{1}{i+j}$ we know that the determinant is given by
$$
\det(A)=\frac{\prod_{i=2}^n\prod_{j=1}^{i-1}(i+j)(-j-i)}{\prod_{i=1}^n\prod_{j=1}^{n}(i+j)}\neq 0.
$$
Hence the matrix has full rank, namely $rank(A)=n$.
The matrix represents a positive definite form on a vector space of dim n.;the matrix is non degenerate; is of full rank