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

I am currently working my way through Hoffman&Kunze Linear Algebra. I noticed that a question on this site concerning the exactly same problem (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.

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.

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  • $\begingroup$ This question has already got an excellent answer by @CarloBeenakker, but, for future reference, the tools that you want not to be used indicate that such a question might be more appropriate for our sister site MSE than here. $\endgroup$
    – LSpice
    Commented Jul 9 at 16:43

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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:

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

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