Vandermonde Matrix - Solving a 3x3 Determinant Question

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In summary, the conversation was about a question regarding the 3x3 Vandermonde Matrix and how to take its determinant. The solution was found on Wikipedia using a simple formula.
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obiwan
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[SOLVED] Vandermonde Matrix - Help!

Hi everyone, I am new to the forum.

I'm bit confused in a particular 3by3 Vandermonde Matrix question and really appreciate if someone can help me with it.

Homework Statement

The question states 'Show that the 3 by 3 Vandermonde Determinant:
1 1 1
det x y z = (x-y) (y-z) (z-x)
x^2 y^2 z^2

I have not made an attemt to the question as I haven't got a clue.

Your help and time are much appreciated. thanksPS. Just realized the rows are not aligned under one another but It is a 3by3 Matrix with the first rows 1, 1, 1 second Row x, y, z and the third row x^2, y^2, z^2.
 
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  • #2
Well, how do you take the determinant of a 3x3 matrix?
 
  • #3
Thanks Cristo. I've managed to show the determinant using a simple formula I came across on wikipedia. Thanks for your help anyway.
 

1. What is a Vandermonde matrix?

A Vandermonde matrix is a square matrix with a specific form in which the elements of each row are increasing powers of a variable. It is named after mathematician Alexandre-Théophile Vandermonde.

2. What is the purpose of a Vandermonde matrix?

The main purpose of a Vandermonde matrix is to simplify polynomial interpolation and polynomial evaluation. It also has applications in signal processing, coding theory, and cryptography.

3. How do you create a Vandermonde matrix?

To create a Vandermonde matrix, you start by choosing a set of distinct values for the variable and then raise each value to the powers of 0, 1, 2, and so on, depending on the size of the matrix. The resulting values are then arranged in the matrix in the form of rows or columns.

4. What is the determinant of a Vandermonde matrix?

The determinant of a Vandermonde matrix is the product of the differences between all possible pairs of the chosen values for the variable. It can also be written as the product of the factorials of the matrix size minus one.

5. How is a Vandermonde matrix used in polynomial interpolation?

A Vandermonde matrix is used in polynomial interpolation by setting up a system of linear equations with the known points on a polynomial curve. The matrix is then used to solve for the coefficients of the polynomial, which can then be used to find the value of any unknown point on the curve.

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