Computing examples without memorizing a long formula

Practical strategy: track basis images $T\hat{\mathbf{i}},T\hat{\mathbf{j}},T\hat{\mathbf{k}}$ and expand cross products in that basis . Geometry first; symbolic algebra handles 3×3 cases quickly.

If $T$ scales the $z$-axis by $3$ and fixes $x,y$, then $T=\begin{bmatrix}1&0&0\\0&1&0\\0&0&3\end{bmatrix}$ and $\det(T)=3$. Uniform scaling by $s$ on all coordinates multiplies every cross magnitude by $s^2$ because each argument scales by $s$.

Write $T\mathbf{a}$ and $T\mathbf{b}$ explicitly in the standard basis when learning; the algebra is manageable in three dimensions and builds intuition faster than a black-box formula.

Shear maps with determinant one preserve volume factor (unimodular) even though they distort angles and lengths. Cross length need not be preserved even when $\det(T)=1$ unless $T$ is a rotation.

If $T$ doubles every coordinate, $\|\mathbf{a}\times\mathbf{b}\|$ changes by factor $4$ from bilinearity: each vector length doubles, and cross magnitude scales as the product of the two scaling factors on the arguments.

Shear with $\det(T)=1$ can distort angles while preserving volume. Cross length may change even though the volume factor stayed at one.