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.
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.