Vandermonde Determinant for NxN Matrices

[grassstrip1]

The problem I have is this: Show that

\begin{bmatrix} 1 & 1 & 1 \\ λ_{1} & λ_{2} & λ_{3} \\ λ_{1}^{2} & λ_{2}^{2} & λ_{3}^{2} \end{bmatrix}
Has determinant

$$ (λ_{3} - λ_{2}) (λ_{3} - λ_{1}) (λ_{2} - λ_{1}) $$

And generalize to the NxN case (proof not needed)Obviously solving the 3x3 was not hard, I simply expanded the expression for the determinate given and showed it to be the same as the one i calculated using the rule of Sarrus.

However for the nxn case I'm not sure how to proceed. I tried entering bigger and bigger matrices into wolfram but there was no clear pattern. Is there a non computational way to solve this?

 

[fresh_42]

Have you tried an induction over $n$?

 

[Charles Link]

The problem I have is this: Show that

\begin{bmatrix} 1 & 1 & 1 \\ λ_{1} & λ_{2} & λ_{3} \\ λ_{1}^{2} & λ_{2}^{2} & λ_{3}^{2} \end{bmatrix}
Has determinant

$$ (λ_{3} - λ_{2}) (λ_{3} - λ_{1}) (λ_{2} - λ_{1}) $$

And generalize to the NxN case (proof not needed)Obviously solving the 3x3 was not hard, I simply expanded the expression for the determinate given and showed it to be the same as the one i calculated using the rule of Sarrus.

However for the nxn case I'm not sure how to proceed. I tried entering bigger and bigger matrices into wolfram but there was no clear pattern. Is there a non computational way to solve this?

I don't have a complete answer, but a 4x4 with the 4th row being cubic will result in a 6th power polynomial form. The number of combinations of 4 things taken 2 at a time is 6, making for the 6 factors. (I don't know how to choose the minus or plus sign for the terms, but if it does generalize, this might be helpful.)...editing... One additional observation is if two columns of a matrix are identical, I think the determinant is zero. This is consistent with setting the product of the factors equal to zero.

 

[LCKurtz]

Here is a link to an old thread about this:
https://www.physicsforums.com/threa...for-a-linear-combination.755660/#post-4759303

 

[grassstrip1]

Thank you! The formula for the vandermonde determinant was exactly what I needed!