Adjugate / cofactor matrix viewpoint (conceptual)

The adjugate (classical adjugate) records how normals and oriented 2-faces transform under a linear map. Conceptually, $T(\mathbf{a}\times\mathbf{b})$ relates to $T\mathbf{a}\times T\mathbf{b}$ with a factor involving $\det(T)$ for invertible $T$ . You need not memorize adjugate formulas to retain the geometric warning.

If $\det(T)=0$, the map collapses three-dimensional volume to a subspace. Cross outputs may still be nonzero for some inputs, but the image lies in a flattened region where 3D volume data is lost.

Think of squeezing a solid to a pancake: many distinct cross inputs can map to similar outputs, so transformed crosses carry less reliable orientation information.

Composition $S\circ T$ multiplies determinants because volume scales multiply along chains of maps. This multiplicativity is why orientation bookkeeping composes cleanly even when cross interchange fails.

Example question: if $T$ collapses $\mathbb{R}^3$ to a plane, can $T(\mathbf{a}\times\mathbf{b})$ be nonzero? Yes, when $\mathbf{a}\times\mathbf{b}$ has a component that maps outside the kernel, but the result still lies in that plane; many crosses collapse depending on the vectors.

You need not memorize adjugate formulas to retain the geometric warning: when volume collapses, cross-based normals become untrustworthy without extra normalization.