- #1
jostpuur
- 2,116
- 19
Anyone having any ideas about how this formula could be proved?
[tex]
\underset{k<l}{\prod_{k,l=1}^n} (x_k - x_l) = (-1)^{n(n-1)/2} \left|\begin{array}{ccc}
1 & \cdots & 1 \\
x_1 & \cdots & x_n \\
\vdots & & \vdots \\
x_1^{n-1} & \cdots & x_n^{n-1} \\
\end{array}\right|
[/tex]
I tried induction without success. The induction step seems to only require other difficult formulas.
[tex]
\underset{k<l}{\prod_{k,l=1}^n} (x_k - x_l) = (-1)^{n(n-1)/2} \left|\begin{array}{ccc}
1 & \cdots & 1 \\
x_1 & \cdots & x_n \\
\vdots & & \vdots \\
x_1^{n-1} & \cdots & x_n^{n-1} \\
\end{array}\right|
[/tex]
I tried induction without success. The induction step seems to only require other difficult formulas.