<s>[INST] How can I prove that this matrix is nonsingular? How can I prove that $
\begin{vmatrix} 
\frac{1}{(x_1-y_1)^2} & \frac{1}{(x_1-y_2)^2} & \cdots & \frac{1}{(x_1-y_n)^2} \\
\frac{1}{(x_2-y_1)^2} & \frac{1}{(x_2-y_2)^2} & \cdots & \frac{1}{(x_2-y_n)^2} \\
\vdots & \vdots & \ddots & \vdots \\
\frac{1}{(x_n-y_1)^2} & \frac{1}{(x_n-y_2)^2} & \cdots & \frac{1}{(x_n-y_n)^2}
\end{vmatrix} \ne 0
$?
For $x_i$ and $y_i$ are all distinct.
 [/INST] Even if all the $x_i$ and $y_j$ are distinct, this is not true. 
For instance, if $x_1=2/3$, $x_2=2$, $y_1=1$, $y_2=0$, we have
$$
\begin{vmatrix}\frac1{(x_1-y_1)^2}&\frac1{(x_1-y_2)^2}\\\frac1{(x_2-y_1)^2}&\frac1{(x_2-y_2)^2}\end{vmatrix}
=\begin{vmatrix}
\frac1{(2/3-1)^2}&\frac1{(2/3-0)^2}\\
\frac1{(2-1)^2}&\frac1{(2-0)^2}
\end{vmatrix}
=\begin{vmatrix}
9&9/4\\
1&1/4
\end{vmatrix}=0
$$
</s>