Cauchy determinant formula

Main Result This note is about Cauchy matrices of the form $$ M = M(\mathbf{a}, \mathbf{b}) = \left( \frac{1}{a_i - b_j} \right)_{1 \leq i, j \leq n} $$ where $\mathbf{a}, \mathbf{b} \in \mathbb{C}^n$.