This question is related to, and a follow-up for, this question. The notation here follows that of the text quoted there.
Let $\matrix M_n$ be a Vandermonde matrix of size $n$ by $n$. The columns of $\matrix M$ are orthogonal to each other, therefore each column is mapped onto a coordinate axis. $w$ is a $n^{th}$ root of unity, and $n$ is a power of $2$. The inversion formula is supposedly:
$\matrix M_n(w)^{-1} =\frac{1}{n}\matrix M_n\left(w^{-1}\right)$
The book explains why multiplying one column by another results in 1. Therefore I understand why multiplying $M(w)$ by its complex conjugate results in $1$. What is $\frac{1}{n}$ though? Why is it part of the equation and what is it?