You are being dismissive of Axler (with the "personal crusade" comment) and disagree with his position, but it would be less disrespectful of Axler and more beneficial to the discussion (and interesting to me) to give at least some argumentation to match Axler's. That is, why is Axler's approach (in the short paper or the book) worse than the traditional approach?
From my skimming of that article, Axler's main complaint with determinants is that it pedagogically leaves students with the impression that eigenvalues are somehow a property of the determinant rather than being a fundamental property of a linear transformation.
I'm not sure I've ever met anyone who has that view; in my experience, more people leave linear algebra with a laundry list of matrix properties whose utility isn't obvious, with determinants and eigenvalues both on that list. This in general is a pedagogical issue with teaching linear algebra as plug-and-chug techniques without explaining why we're doing them.
Truthfully, I'm not convinced that introducing the determinant as "just" the product of eigenvalues is itself useful. One of the more useful properties of the determinant is that it is built up via row-reduction, and consequently, it can be computed using Gaussian elimination. I've seen some texts actually define the determinant via its row-reduction properties, and then build up to demonstrating the other formulas for it.
I'll also point out that there are two more uses that Axler doesn't acknowledge: the determinant tells you the volume of a parallelpiped, and it's a convenient way to remember the formula for a cross product.
It's not. For all I care he may write his book without introducing determinants or only introducing them at the very end. I'm all for different pedagogical approaches.
But he writes a paper called "down with determinants" and starts it with "if you think complex matrices have an eigenvalue because the characteristic polynomial has a root, then this is wrong". But it's not wrong, mathematically it's 100% correct! It's just his own personal opinion because he somehow doesn't like determinants.
Mathematics is all about different approaches and different tools, not about "the one true enlightened way", as he makes it out to be. This is why I'm calling it a "personal crusade". Had he just called his paper "a determinant-free approach to linear algebra" I would take no issue.
The degree to which classrooms and textbooks have converged onto Axler's pedagogical approach is the degree to which Axler's position is "personal" vs mainstream.
