The Cauchy matrix has the useful properties that all square Cauchy matrices are invertible and every submatrix of a Cauchy matrix is also a Cauchy matrix. Used together with the useful properties of the Galois field make the Cauchy matrix extremely useful for erasure coding.

With two sequences x and y, which only contain distinctive elements, we can get the Cauchy matrix with the following formula:

If we strictly use a Cauchy matrix without augmenting it, for example with the identity matrix to get a systematic code, we have the following formula to directly get the inverse, without having to resort to decomposition methods:

Formula above taken from:

The Art of Computer Programming
Volume 1: Fundamental Algorithms
Donald E. Knuth - 1997
Paragraph 1.2.3: Sums and Products
Page 38, Exercise 41