Vandermonde Matrix Identity for Cauchy Matrix/Examples/3x3

Example of Vandermonde Matrix Identity for Cauchy Matrix
Illustrate $3 \times 3$ case for Vandermonde Matrix Identity for Cauchy Matrix and Value of Cauchy Determinant.

Let $C$ denote the Cauchy matrix of order $3$:


 * $C = \begin {pmatrix}

\dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \dfrac 1 {x_1 - y_3} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \dfrac 1 {x_2 - y_3} \\ \dfrac 1 {x_3 - y_1} & \dfrac 1 {x_3 - y_2} & \dfrac 1 {x_3 - y_3} \\ \end{pmatrix}$

where the values in $\set {x_1, x_2, x_3, y_1, y_2, y_3}$ are assumed to be distinct.

Then:

Proof
Define Vandermonde matrices

Define polynomials:

Define invertible diagonal matrices:

Then:

Determinant of Diagonal Matrix gives:

Value of Vandermonde Determinant gives:

Determinant of Matrix Product and Definition:Inverse Matrix give:

Then:

Insert the four determinant equations and simplify to obtain the equation for $3 \times 3$ $\map \det C$.