A linear endomorphism of $\mathbb R^n$ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant

Suppose first that $A$ is invertible. For every elementary matrix $E_0$, both $E_0$ and $E_0^{-1}$ are Lipschitz by [L5], so $\partial(E_0F)=E_0(\partial F)$ for every bounded set $F$. Thus [L4] makes $E_0F$ Jordan whenever $F$ is Jordan. Coordinate permutations and nonzero coordinate scalings send rectangular figures to rectangular figures, with content factor $1$ and $|c|$ respectively; applying this to arbitrarily close inner and outer figures from [L8] proves those factors for every bounded Jordan $F$. For a shear adding $c$ times one coordinate to another, take rectangular figures $P\subseteq F\subseteq Q$ from [L8] with $\operatorname{cont}(Q)-\operatorname{cont}(P)$ arbitrarily small. Sections of a rectangular figure parallel to the changed coordinate are finite unions of intervals, hence Jordan at every parameter, and the corresponding sections of $E_0P$ and $E_0Q$ are their translates by a quantity depending only on the fixed coordinates, so they are finite unions of intervals of the same total length. Both hypotheses of [L3] are therefore met by $P,Q,E_0P,E_0Q$ but are not claimed for $F$, whose sections need not be Jordan; [L3] gives $\operatorname{cont}(E_0P)=\operatorname{cont}(P)$ and $\operatorname{cont}(E_0Q)=\operatorname{cont}(Q)$. Since $E_0P\subseteq E_0F\subseteq E_0Q$ and $E_0F$ is already known Jordan, its content and that of $F$ are both squeezed between $\operatorname{cont}(P)$ and $\operatorname{cont}(Q)$, so a shear preserves content. These are exactly the absolute determinant factors listed by [L2].

By [L1], write $A$ as a finite product of elementary matrices. Apply step 1.1 successively to $E$: every intermediate image is bounded Jordan, and its content is multiplied by the corresponding absolute determinant factor. The row-operation laws [L2], applied successively from the identity, identify the product of those factors with $|\det A|$. Boundedness follows from [L5].

If $A$ is singular, [L6] reduces it to an echelon matrix with a zero row; [L7] realizes this as an invertible change of codomain coordinates. The transformed range lies in a coordinate hyperplane, whose bounded part fits in slabs of arbitrarily small thickness and therefore has content zero and is Jordan. Applying the invertible case to undo the coordinate change shows that $T(E)$ has content zero and is Jordan; the real criterion in [L1] gives $\det A=0$, so the same formula holds.