I want to prove that the following 2 matrices have the same determinant (which they do and it is equal to 288) Apologies for quality: [Matrix one][1] [Matrix two][2]
I know that the first matrix is a Vandermonde matrix, and it is easy to calculate its determinant. The second matrix looks similar, but I can't find the connection. Any ideas would be helpful.
Let $A$ be the first matrix, and $B$ the second matrix , then
$ B = A C $
where
$ C = \begin{bmatrix} 1 && -1 && 0 && 0 && 0 \\0 && 1 && -1 && 0 && 0 \\ 0 && 0 && 1 && -1 && 0 \\ 0 && 0 && 0 && 1 && -1 \\ 0 && 0 && 0 && 0 && 1 \end{bmatrix} $
From the well-known property of determinants we know that
$ \det(B)= \det(A) \det(C) $
Since $C$ is upper-triangular, then its determinant is the product of its diagonal elements, which is $1$, hence
$ \det(B) = \det(A) $