Explaining why the determinant measures the change of volume under a matrix

Question

I would like to explain to my students that the determinant of a matrix $A$ is in fact the measure of how the volume changes under said matrix. To do that, I would like to start from the $n$-dim. cube (represented by the standard basis $e_1,\ldots,e_n$ of $\mathbb{R}^n$) and derive the formula for the volume of the parallelepiped determined by $Ae_1,\ldots,Ae_n$. Of course, I could go to the exterior algebra $\Lambda^n(\mathbb{R}^n)$ and work there, but I don't think this would be very pedagogical...

Thus, my question is: is there a down-to-earth way to recover the determinant of $A$ as the volume of the parallelepiped defined above? And if not, how can one make a compelling argument of why taking the exterior product $Ae_1\wedge\ldots\wedge Ae_n$ gives the volume?

Answer 1

My preferred pedagogical approach is to start with examples in $2$ and $3$ dimensions. In $2D$ we can easily see that the columns of the matrix $A$ are the vectors $\vec u= A\vec i$ and $\vec v=A \vec j$ ( where $\{\vec i, \vec j \}$ is the standard basis) and it is easy to show that the ''signed'' area of the parallelogram with sides $\vec u$ and $\vec v$ is the cross product $ \vec u \times \vec v = \det A$.

In a similar way we can show that the mixed product of the vectors $\vec u=A\vec i$,$\vec v=A \vec j$ and $\vec w=A\vec k$ is the volume of the of the transformed parallelepiped,and these vectors are the columns of the matrix $A$ so that $\vec u\cdot(\vec v \times \vec w)=\det A$.

Now the generalization to $n-$dimensional space can be done noting the properties of the determinant, especially the multi-linearity, and noting that we can go from the volume of a $n-$cube to an $n-$polyhedron by linear tranformations of the sides.

Answer 2

In 2-dimension, if you slide the side $AB$ for a parallelogram $ABCD$ along the line containing $A$ and $B$, the area remains the same. To compute the area of parallelogram $ABCD$, one can apply a shear operation to $ABCD$ and turn it into a rectangle first.

Using this sort of operation, one can show the signed-area of $ABCD$, as a function of vectors $\vec{AB}$ and $\vec{AC}$, is anti-symmetric and multi-linear. Now invite the student to express all these into linear algebra, the formula for determinant for $2 \times 2$ matrix should come up naturally.

Next, generalize this to 3-dimension. As an exercise, ask the students to compare this way of computing volume by "shearing" with how one computing determinant by elementary row (or column?) operations.