Can the theory of determinants be derived using the definition by row operations?

Question

This year I'm teaching an elementary course on linear algebra for physics students. Because of that I have been researching the different ways of presenting the definition of the determinant (one of the hardest topics for such a course, in my opinion).

In the lecture notes by Terry Loring a definition using elementary row operations is given

http://www.math.unm.edu/~loring/links/linear_s06/det_def.pdf

I like this definition since seems to me very algorithmic, and uses an idea which is familiar to the students. Compare this with the other more popular definitions which are

• The explicit formula using the sign of permutations
• The recursive formula using the Laplace development
• The axiomatic one as the unique multilinear alternate form by rows (or by columns) that takes the value 1 at the identity matrix.

Of course, this definition by row elimination seems very close to this last axiomatic definition.

However my question is, do you know if there is some way of deriving the whole theory of determinants from the definition based on row elimination? Indeed, it even seems hard to prove directly that the definition is correct (non ambiguous).

Do you know some book using this approach?

Answer 1

If I remember correctly, Apostol's Calculus Vol. 2 approaches it the way that egreg outlined in his comment. I could check the table of contents on Amazon and verify that the title of Chapter 3 is "Determinants".

I don't have my copy to hand, and it's also possible that instead he gives the axioms requiring it to be a multilinear form with some other behaviors and then shows that row elimination is an effective way to compute it -- it's been a while since I read it. But either way I'll suggest you check it out.

Answer 2

I also liked very much the approach Ted Shifrin used to introduce determinants. He motivated it by finding the inverse matrix of a $2 \times 2$ matrix and then interpreting the factor $ad-bc$ which shows up just by using some vector algebra and scalar products. He then showed that the determinant resembles the signed area of the parallelogram in 2D and signed volume of the parallelopiped in 3D which has certain properties. Later he used those properties to motivate the abstract definition in $n$ dimensions.

His book is called Multivariable Mathematics, but there is also a playlist called "Shifrin Math 3500" on YouTube where you can watch his lectures for free. Generally his way of motivating Linear Algebra concepts are the best ones I have seen so far.

Answer 3

I finally have found what I wnated in this nice article

JOURNAL ARTICLE A Fresh(man) Treatment of Determinants Kenneth P. Bogart The American Mathematical Monthly Vol. 96, No. 10 (Dec., 1989), pp. 915-920

http://www.jstor.org/stable/2324588

Indeed, Bogart takes the action of elementary rows operations on determinants as axioms (as well as the fact that the determinant of the identity is 1). And shows that the determinant function exists using the Laplace expansion and induction.

Many thanks everyone for their comments!