16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 1.6.12

Systems of Linear Equations

Mathematics — Linear Algebra · hilbert-matrix-invertibility

Problem

The result of Example 16 suggests that perhaps the matrix

A = \begin{bmatrix} 1 & \frac{1}{2} & \cdots & \frac{1}{n} \\ \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n+1} \\ \vdots & \vdots & & \vdots \\ \frac{1}{n} & \frac{1}{n+1} & \cdots & \frac{1}{2n-1} \end{bmatrix}

is invertible and

A^{-1}

has integer entries. Can you prove that?