# An application of linear algebra in Calculus

Definition. For an integer $n ge 1,$ the $ntimes n$ Hilbert matrix is defined by $H_n=[a_{ij}],$ where $displaystyle a_{ij}=frac{1}{i+j-1}, 1 le i,j le n.$ It is known that $H_n$ is invertible and if $H_n^{-1}=[b_{ij}],$ then $displaystyle sum_{i,j}b_{ij}=n^2.$ We are going to use these two properties of…