I am currently working my way through Hoffman&Kunze Linear Algebra. I noticed that a question on this site (here: Deriving inverse of Hilbert matrix)has been asked about proving that the entries of inverse of the hilbert matrices are integers that avoids the explicit computation of it, so the derivation of the explicit formula isn't concerned.

And my question is, how to derive the formula for $H^{-1}$, in which $H_{ij}=\frac{1}{i+j-1}$

I notice there is an essay On the Inversion of Certain Matrices by Samuel Schechter

But I really can't make my way through the proof. So can somebody show me or guide me to better(more modern) resources for the way to prove in a simpler way than using lagrangian interpolations？(I haven't learned it, but it would also be great if you can unrip the mist of it inside the proof)

PS: It would be greater if your answer is related to the Cauchy Matrices, supposedly a generalization of Hilbert Matrices