Let: {{begin-eqn}} {{eqn | l = C       | r = \begin{bmatrix} \dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\ \end{bmatrix} }} {{end-eqn}} To be proved: {{begin-eqn}} {{eqn | l = \det \paren {C}        | r = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_i - y_j}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i - y_j}\right)}       | c = Knuth (1997) replacing $y_k \to -y_k$ in $C$ and $\det \paren {C}$ }} {{end-eqn}} Assume hereafter that set $\set {x_1,\ldots,x_n,y_1,\ldots,y_n}$ consists of distinct values, because otherwise $\det \paren {C}$ is undefined or zero. '''Preliminaries''': Vandermonde Matrix Identity for Cauchy Matrix supplies matrix equation :$\displaystyle (1)\quad - C = PV_x^{-1} V_y Q^{-1}$ :Definitions of symbols: ::$\displaystyle V_x=\paren {\begin{smallmatrix} 1         & 1         & \cdots & 1 \\ x_1       & x_2       & \cdots & x_n \\ \vdots    & \vdots    & \ddots & \vdots \\ x_1^{n-1} & x_2^{n-1} & \cdots & x_n^{n-1} \\ \end{smallmatrix} },\quad  V_y=\paren {\begin{smallmatrix} 1         & 1         & \cdots & 1 \\ y_1       & y_2       & \cdots & y_n \\ \vdots    & \vdots    & \ddots & \vdots \\ y_1^{n-1} & y_2^{n-1} & \cdots & y_n^{n-1} \\ \end{smallmatrix} }$ Vandermonde matrices ::$\displaystyle P= \paren {\begin{smallmatrix} p_1(x_1) &  \cdots & 0 \\ \vdots   & \ddots  & \vdots \\ 0        & \cdots  & p_n(x_n) \\ \end{smallmatrix} }, \quad Q= \paren {\begin{smallmatrix} p(y_1)  & \cdots  & 0 \\ \vdots  & \ddots  & \vdots \\ 0       & \cdots  & p(y_n) \\ \end{smallmatrix} }$  Diagonal matrices ::$\displaystyle  p(x) = \prod_{i \mathop = 1}^n \paren {x - x_i}, \quad \displaystyle  p_k(x) = \prod_{i \mathop = 1,i \mathop \ne k}^n \, \paren {x - x_i}, \quad 1 \mathop \le k \mathop \le n$ Polynomials '''Determinant of $C$ Calculation''': {{begin-eqn}} {{eqn | l = C       | r = \paren {-I}\,P\,V_x^{-1}\, V_y\, Q^{-1}       | c = Vandermonde Matrix Identity for Cauchy Matrix  Symbol $I$ is the $n\times n$ identity matrix. }} {{eqn | l = (2)\quad  \det \paren {C}         | r = \det \paren { -I } \det \paren {P} \det \paren { V_x^{-1} } \det \paren { V_y } \det \paren { Q^{-1} }        | c = Determinant of Matrix Product }} {{eqn | r =   \paren { -1 }^n  \det \paren { I } \det \paren {P} \det \paren { V_x^{-1} } \det \paren { V_y } \det \paren { Q^{-1} }      | c = Effect of Elementary Row Operations on Determinant  Factor constant $-1$ from all rows of $-I$.  }} {{eqn | r =   \paren { -1 }^n  \det \paren { I } \dfrac { \det \paren {P}  \det \paren { V_y } } {  \det \paren { Q } \det \paren { V_x } }       | c = Matrix Product with Adjugate Matrix and Determinant of Matrix Product }} {{end-eqn}} '''Lemma''': $\det \paren {P} = \paren {-1}^m \paren { \det \paren {V_x} }^2$ where $m=\frac 1 2 n \paren {n-1}$ :'''Details''': Determinant $\det \paren {P}$ expands to: {{begin-eqn}} {{eqn | l = \,\,\prod_{j \mathop = 1}^n \map {p_j} {x_j}        | r =  \prod_{j \mathop = 1}^n \,\,  \prod_{k \mathop = 1,\, k \mathop \neq j}^n \paren { x_j - x_k }       | c = Definition of polynomials $\map {p_j} x$. }} {{end-eqn}} :Pair factors $\paren {x_r - x_s}$ and  $\paren {x_s - x_r}$ into factor $- \paren {x_s - x_r}^2$, then: {{begin-eqn}} {{eqn | l = \,\,\det \paren {P}        | r =   \paren {-1}^m  \,  \paren { \prod_{1 \mathop \leq r \mathop \lt s \mathop \le n} \paren { x_s - x_r }^2 }       | c = where $m = 1 + \cdots + \paren {n-1} = \frac 1 2 n \paren {n-1}$ }} {{eqn | r = \paren {-1}^m  \,  \paren { \det \paren {V_x} }^2       | c = Vandermonde Determinant }} {{end-eqn}} {{qed|lemma}} Apply the '''Lemma''' to equation (2): {{begin-eqn}} {{eqn | l = \det \paren {C}        | r =   \paren { -1 }^{n+m}  \, \paren { \dfrac { \det \paren {V_x}  \det \paren { V_y } } {  \det \paren { Q } } }       | c = $m = \frac 1 2 n \paren {n-1}$ }} {{eqn | r =   \paren { -1 }^{n+m}   \,\, \dfrac { \displaystyle   \prod_{1 \mathop \le m \mathop \lt k \mathop \le n}^{\phantom n} \paren { x_k - x_m  }   \,   \prod_{1 \mathop \le m \mathop \lt k \mathop \le n} \paren { y_k - y_m  }  } { \displaystyle     \prod_{k \mathop = 1}^n \map {p} {y_k}  }     | c = Vandermonde Determinant and Determinant of Diagonal Matrix }} {{eqn | r = \paren { -1 }^{n+m}   \,\, \dfrac { \displaystyle   \prod_{1 \mathop \le m \mathop \lt k \mathop \le n}^{\phantom n} \paren { x_k - x_m  } \paren { y_k - y_m  }  } { \displaystyle     \prod_{k \mathop = 1}^n \prod_{j \mathop = 1}^n \paren { y_k - x_j } }     | c = Definition of $\map p x$. }} {{eqn | r =   \paren { -1 }^{n+m}   \,\, \dfrac { \displaystyle   \paren {-1}^{m}\prod_{1 \mathop \le m \mathop \lt k \mathop \le n}^{\phantom n} \paren { x_k - x_m  } \paren { y_m - y_k }  } { \displaystyle     \paren {-1}^{n^2}\,\,\prod_{k \mathop = 1}^n \prod_{j \mathop = 1}^n \paren { x_j - y_k } }     | c = Factor out changed signs. }} {{eqn | r =  \left. \dfrac { \displaystyle   \prod_{1 \mathop \le m \mathop \lt k \mathop \le n}^{\phantom n} \paren { x_k - x_m  } \paren { y_m - y_k }  } { \displaystyle     \prod_{k \mathop = 1}^n \prod_{j \mathop = 1}^n \paren { x_j - y_k } } \right.     | c = All signs cancel. }} {{end-eqn}} {{qed}}