# Vandermonde determinant from 3-by-3

I am a novice at math and need help calculating the determinant of the Vandermonde matrix, which is:

$$V=\begin{bmatrix} 1&a_1&a_1^2&\dots&a_1^{n-1}\\ 1&a_2&a_2^2&\dots&a_2^{n-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 1&a_n&a_n^2&\dots&a_n^{n-1} \end{bmatrix}$$

$$det(V)={\displaystyle \prod_{1\le i\lt j\le n} (a_j-a_i)}$$

I want to start with the following 3-by-3 and generalize to the n-by-n.

$$A=\begin{bmatrix} 1&a_1&a_1^2\\ 1&a_2&a_2^2\\ 1&a_3&a_3^2 \end{bmatrix}$$

So far I have:

$$det(A) = a_2a_3(a_3-a_2)-a_1a_3(a_3-a_1)+a_1a_2(a_2-a_1)$$

But I cannot figure out how to simplify this into a form like the determinant for $$V$$ above. What do I need to do?

• Hint: if $a_i = a_j$ then there are two identical rows in the matrix, so the determinant is 0. By the polynomial factor theorem, $a_j - a_i$ is a factor of the determinant. Commented Feb 27, 2021 at 2:49
• One way to get started is to do row operations on $A$. In particular, try subtracting the first row from the others, and consider what factors would be common to the new rows. Commented Feb 27, 2021 at 2:51
• It is not clear to me what you are asking. You already know the complete factorization of the determinant into linear factors. What do you want to do now? Prove that the factorization is equal to the determinant? It is easy enough to completely expand the product and also the determinant and verify that they are equal. Commented Feb 27, 2021 at 3:39
• Wikipedia has an article on this: Vandermonde matrix Commented Feb 27, 2021 at 10:09

## 1 Answer

I am a novice as well. I tried figuring this out.

$$\begin{bmatrix}1 & a& a^2 \\1&b&b^2 \\1 &c& c^2 \end{bmatrix}$$ $$R_2 \rightarrow R_2 -R_1 \\ R_3 \rightarrow R_3 -R_1$$ $$\begin{bmatrix}1 & a& a^2 \\0&b-a&b^2-a^2 \\0 &c-a& c^2-a^2 \end{bmatrix}$$ Taking out $$(b-a)(c-a)$$ from rows $$R_2,~ R_3$$ $$(b-a)(c-a) \cdot \begin{bmatrix}1 & a& a^2 \\0&1&b+a \\0 &1& c+a \end{bmatrix}$$ Taking Determinant gets us. $$|A| = (b-a)(c-a)(c-b)$$

• When taking out $(b-a)(c-a)$ you dont take it out of the matrix but out of the determinante, apart from that your answer seems good. Commented Mar 2, 2023 at 7:09