The entry sum of the inverse Cauchy matrix

Let $x_1,x_2,\dots ,x_n$ be $n$ numbers, and $y_1,y_2,\dots ,y_n$ be $n$ further numbers chosen such that all $n^2$ pairwise sums $x_i+y_j$ are nonzero. Consider the $n\times n$-matrix $$C:={\left({\displaystyle \frac{1}{x_i+y_j}}\right)}_{1\le i\le n,1\le j\le n}=\left(\begin{array}{cccc}{\displaystyle \frac{1}{x_1+y_1}}& {\displaystyle \frac{1}{x_1+y_2}}& \cdots & {\displaystyle \frac{1}{x_1+y_n}}\\ {\displaystyle \frac{1}{x_2+y_1}}& {\displaystyle \frac{1}{x_2+y_2}}& \cdots & {\displaystyle \frac{1}{x_2+y_n}}\\ \mathrm{⋮}& \mathrm{⋮}& \ddots & \mathrm{⋮}\\ {\displaystyle \frac{1}{x_n+y_1}}& {\displaystyle \frac{1}{x_n+y_2}}& \cdots & {\displaystyle \frac{1}{x_n+y_n}}\end{array}\right)\mathrm{.}$$ This matrix $C$ is known as the “Cauchy matrix”, and has been studied for 180 years. A classical result says that if $C$ is invertible, then the sum of all entries of its inverse $C^{-1}$ is ${\sum }_{k=1}^n{x}_k+{\sum }_{k=1}^n{y}_k$. We give a simple and short proof of this result, and briefly discuss a “tropicalized” variant in which the entries ${\displaystyle \frac{1}{x_i+y_j}}$ are replaced by $\min \{x_i,y_j\}$.