Determinant defined as Product of Columns

Question

Let $N$ be a non-singular matrix, $v_i$ be the column $i$ of $N$, and $M$ be a matrix with $e_i$ as columns.

$M$ and $N$ have the same dimensions.

I do not understand how

$|\det(N)|=\prod_i ||v_i||\cdot |\det(M)|$.

This is part of the proof of Hadamard's inequality on Wikipedia.

Answer 1

Suppose $$N = [l_1v_1 \ l_2v_2 \ \ldots l_nv_n]$$ where $l_i=||v_i||$ Then $$detN = \prod_il_i \cdot det[v_1 \ v_2 \ \ldots v_n] = \prod_il_i \cdot detM$$ due to determinant's linearity with respect to each column.

Adding $||\ ||$, we get the desired result.