A coordinate scaling and a coordinate transposition send the unit cube to a set of measure equal to the absolute value of the determinant

Given: A natural number $n\ge 1$, the Axiom of Countable Choice, and the elementary matrices $D_p(c)$ and $E_{pq}$ over $\mathbb{R}$.

If $a_i\le b_i$ are real for $i<n$, then any box obtained from the coordinate interval product ${\prod }_{i<n}[a_i,b_i]$ by independently choosing for each endpoint whether it is included has Lebesgue measure ${\prod }_{i<n}(b_i-a_i)$ (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included). In particular $(u,v{]}^n=B(\mathbf{u},\mathbf{v})$ (Half-open boxes in ${\mathbb{R}}^n$ and their volume).

An elementary matrix is a matrix obtained by applying one elementary row operation to the identity matrix $I_n$; there are three types: $E_{pq}$ interchanges rows $p$ and $q$; $D_p(c)$ multiplies row $p$ by $c\mathrm{\ne }0$; and $T_{pq}(c)$ adds $c$ times row $q$ to the distinct row $p$ (Elementary matrices obtained by applying one elementary row operation to an identity matrix, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes).

Let $n\ge 1$ and let $A\in M_n(R)$ be a matrix over a commutative ring; interchanging two rows changes $\det (A)$ to $-\det (A)$, and multiplying one row by any $c\in R$ changes it to $c\det (A)$ (For every square matrix, including singular ones, a row swap negates the determinant, scaling a row by any scalar scales it, and row addition leaves it unchanged, claims 1 and 2; For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix).

If $A$ is upper or lower triangular over a commutative ring, with $n\ge 1$, then $\det (A)={\prod }_{i<n}{a}_{ii}$ (The determinant of a triangular matrix is the product of its diagonal entries).

For every linear $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ there is a unique matrix $A$ such that $(Lh{)}_i={\sum }_{j<m}{a}_{ij}{h}_j$ (Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$).

The absolute value satisfies $\mathrm{\mid }c\mathrm{\mid }>0$ for $c\mathrm{\ne }0$, $\mathrm{\mid }c\mathrm{\mid }=c$ for $c\ge 0$ and $\mathrm{\mid }c\mathrm{\mid }=-c$ for $c\le 0$ (Absolute value in an ordered field, Basic properties of the absolute value).