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I'd like to show the case $n=4$ for the Vandermonde-determinant. It should look like this:

$V_4 := \det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x_1 & x_2 & x_3 & x_4 \\ x_1^2 & x_2^2 & x_3^2 & x_4^2 \\ x_1^3 & x_2^3 & x_3^3 & x_4^3 \end{pmatrix} = (x_4 - x_3)(x_4 - x_2)(x_4 - x_1)(x_3 - x_2)(x_3 - x_2)(x_2 - x_1)$.

I tried calculating the $\det(\dots)$ via the first row. However, it ends up in a very large calculation.. I'm sure there's a better way to show especially this case with $n=4$.

Thank you for any help ;)

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Let $$P(x)=\det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x & x_2 & x_3 & x_4 \\ x^2 & x_2^2 & x_3^2 & x_4^2 \\ x^3 & x_2^3 & x_3^3 & x_4^3 \end{pmatrix}$$ then $P$ is a polynomial with degree $3$ and $x_2,x_3,x_4$ are their roots so $$P(x)=\lambda(x-x_2)(x-x_3)(x-x_4)$$ so the given determinant is $$P(x_1)=\lambda(x_1-x_2)(x_1-x_3)(x_3-x_4)$$ and finaly to figure out the leading coefficient $\lambda$ we have $$\lambda=\begin{pmatrix} 1 & 1 & 1 \\ x_2 & x_3 & x_4 \\ x_2^2 & x_3^2 & x_4^2 \end{pmatrix}:=V_3$$ so by simple induction we have $$\lambda=(x_2-x_3)(x_2-x_4)(x_3-x_4)$$

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You can use series of elementary row operations: $-x_1R_3+R_4$, $-x_1R_2+R_3$, $-x_1R_1-R_2$ to get a matrix that is row equivalent to $V_4$. Since it's elementary operations of type 3, the determinant of this matrix is the same as $V_4$, and it is simpler to calculate its determinant.

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Hint: consider the last row $R_4$, and replace it by $R_4-x_4R_3$, the third row $R_3$ replaced by $R_3-x_4R_2$ and $R_2$ replaced by by $R_2-x_4R_1$. Then after these transformations $$V_4=\det\begin{pmatrix}1&1&1&1\\ x_1-x_4&x_2-x_4&x_3-x_4&0\\ x_1(x_1-x_4)&x_2(x_2-x_4)&x_3(x_3-x_4)&0\\ x_1^2(x_1-x_4)&x_2^2(x_2-x_4)&x_3^2(x_3-x_4)&0 \end{pmatrix},$$ which is easier to handle (expand with respect to the last column, factor out $(x_1-x_4)(x_2-x_4)(x_3-x_4)$ and we have to cumpute $V_3$).

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  • $\begingroup$ Hello once again. I just call $x_1=a, x_2=b, x_3=c, x_4=d$ and after calculating the det and factorizing I get: $-1*[(a-d)(b-d)(c-d)*[bc(c-b)+ab(b-a)+ac(a-c)]]$. Isn't there something wrong ? =/ $\endgroup$ – Vazrael Mar 5 '14 at 8:52
  • $\begingroup$ I've edited. Is it clearer? $\endgroup$ – Davide Giraudo Mar 5 '14 at 9:42

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