[rguzman]

one issue that comes to mind is the nature of the determinant. when one considers the determinant defined by the recursive definition, it seems like a highly contrived object that is difficult to work with (as it is from that definition!). avoiding that confusion requires that a lot more scaffolding be built (ala Axler in the "Done Right" book). either way you have some work: either to untangle the meaning of the weird determinant or get to the place where you can understand the determinant as the product of the eigenvalues.

[cooljoseph]

Do you know why introductory textbooks don't define the determinant in terms of the exterior product? This is how some "real" mathematicians I've talked to define it. It also is more intuitive (in my opinion) to define determinants as "signed volumes" than some sum of products multiplied by signs of cycles.

The product of eigenvalues definition is also somewhat intuitive to me ("How much does the matrix scale vectors in each direction? Now multiply those numbers together."), but it's harder to motivate the fact that adding rows together doesn't change the determinant, which is kind of important to actually computing the determinant.

[myworkinisgood]

Really? You want to teach freshman people exterior products?