It sure is (provided you count multiplicities correctly), but that's not the one-sentence explanation I would have gone with, even to someone with much more intuition about eigenvalues than you'd get from this document. I would say the determinant is the volume scale factor (or hypervolume scale factor, in the general case).
Actually, what's really interesting to me is your general point: why do <teaching materials for mathematics subject X> never mention <the key insights that made subject X clear to me>? And why does this question not get asked more often? The nearest I've been able to come to a plausible answer is a mixture of (i) the key insight varies from person to person more than you'd think, combined with the closely related (ii) you can't teach key insights just by saying a few words.
Thank you for your comment! First off the explanation regarding det(A) = the volume scaling of the transformation T_A associated with the matrix A. I've been stuck for over a week now trying to word this any other way possible because, in the current ordering of the sections, I'm covering determinants before linear transformations.
Perhaps there is no better once sentence than talking about the volume and I should reorder the sections...
You've raised a very important point regarding "routes to concepts" which really should be asked more often!
> (ii) you can't teach key insights just by saying a few words.
Generally true, though we have to say that the key difficulty in communicating insights is lacking prerequisites. Therefore, if you think very carefully about the prerequisite structure (i.e. model of the reader's previous knowledge) you can do lot's of interesting stuff in very few words.
> (i) the key insight varies from person to person more than you'd think,
Let G=(V,E) where V is the set of math concepts, and E are the links between them. Then there are as many ways to click on a concept x, as there there are in-links for x! In this case we have at least three routes:
Route 1: geometric
lin. trans T_A = {matrix:A, B_in:{e1,..,en}, B_out:{f1,..,fn}}
---> what happens to a unit cube 1x1x1,
represented (1,...1) w.r.t. B_in,
after going through T_A?
---> output will be (1,..,1) w.r.t. B_out
---> observe that cols of A are f1..fn
therefore
det(A) = (hyper)volume of (hyper)parallelepiped
formed by vectors f1..fn
Route 2: computational
det(A) = { formula for 2x2, formula for 3x3, ... }
---> a easy test for invertibility of an n by n matrix
sidenote: see Cramer's rule for another application of dets
Route 3: algebraic
given a 2x2 matrix A,
find a coefficients A T D that satisfy the following matrix equation:
B*A^2 + T*A + D = 0,
the linear term is the trace T = sum_i a_ii, the constant term is D = det(A).
sidenote: B*λ^2 + T*λ + D = {characteristic poly. of A} = det(A-λI)
So perhaps the task of teaching math is not so much to try to find "the right" or "the best" route to a concept, but to collect many explanations of routes (edges in G), then come up with a coherent narrative that covers as many edges as possible (while respecting the partial-sorted-order of prerequisites ).
(It's the product of the eigenvalues.)