Pointers to proofs:

• A New Approach to Inversion of a Cauchy Matrix

• Inverse of Cauchy Matrix

For a direct evaluation of the sum of all entries of $H^{-1}$, see The entry sum of the inverse Cauchy matrix

Perhaps the simplest approach first computes the determinant of an arbitrary Cauchy matrix, here is a derivation. Since every submatrix of a Cauchy matrix is also a Cauchy matrix, you can then find the inverse $H$ from the adjoint of $H$.

Some pointers to worked-out calculations:

• A New Approach to Inversion of a Cauchy Matrix

• Inverse of Cauchy Matrix

For a direct evaluation of the sum of all entries of $H^{-1}$, see The entry sum of the inverse Cauchy matrix