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How can you prove the following determinant formula, where the determinant is the same as that of the Cauchy matrix $$\det [ \frac{1}{1 - x_i y_j} ]_{i, j = 1}^n = \frac{\prod_{1 \leq i < j \leq n} (x_i - x_j)(y_i - y_j)}{\prod_{i, j = 1}^n (1 - x_i y_j)}.$$

Jonathan McDonald
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    Can't you do some simple manipulations to the matrix (like multiply the $i$th row by $x_i^{-1}$) to bring it to basically the form of the Cauchy matrix? – Sam Hopkins Nov 04 '23 at 22:14
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    @SamHopkins I think you need to multiply the $i$th row by $x_i$ and then change variables $x_i \mapsto x_i^{-1}$ but yes, this works. – Will Sawin Nov 04 '23 at 22:21
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    Or you may mimic the standard proofs for Cauchy matrix: the determinant is antisymmetric in $x$'s, and also in $y$'s, and has such a denominator as RHS, thus it equals some polynomial times RHS. The degree of this polynomial is 0, so this is a constant. The value of this constant may be found by looking at certain coefficient. – Fedor Petrov Nov 05 '23 at 05:24

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