This article, “A User-Friendly Introduction to the Theory of Determinants” was inspired by a number of discussions of the best way to teach determinants on Facebook. It turns out to be a topic that inspires a lot of opinions from mathematicians.
A warning is in order: “user-friendly” is a relative term. This article is pitched toward undergraduate math majors and above, and even they may find that some assembly is required. It would probably not make sense in an introductory linear algebra course except at an honors level; however, in an honors course which goes on to cover the Cartan-Stokes theorem, it could well be a helpful approach.
It got long and involved enough that the best format seemed to be a PDF. Click here to view it. In my browser (Chrome) it opens in a separate tab, but you may need to download it to view it.