Or, if you're using exact arithmetic (e.g. cryptography), you should use something like the Bareiss algorithm - Gaussian elimination is only polynomial if you use floats: https://en.m.wikipedia.org/wiki/Bareiss_algorithm
The Leibniz formula is totally impractical for calculating determinants, but it is sometimes useful in proofs due to its symmetric form.
I guess if you just throw a definition of the determinant at people and leave it at that, I can understand that it doesn't make sense. But that's not how I was introduced to the concept (I learned about abstract vector spaces and transformations and how they are represented by matrices - and only then about determinants and their properties and different ways of computing them). That's why Axler's critique rings somewhat hollow to me - maybe it's different if you learn about linear algebra mostly in an engineering context as opposed to pure maths.
For theoretical purposes, determinants are important because they constitute a homomorphism between the general linear group and the multiplicative group of the underlying field, and the kernel of this homomorphism is the special linear group. This leads to some very short and elegant proofs.
Or, if you're using exact arithmetic (e.g. cryptography), you should use something like the Bareiss algorithm - Gaussian elimination is only polynomial if you use floats: https://en.m.wikipedia.org/wiki/Bareiss_algorithm
The Leibniz formula is totally impractical for calculating determinants, but it is sometimes useful in proofs due to its symmetric form.
I guess if you just throw a definition of the determinant at people and leave it at that, I can understand that it doesn't make sense. But that's not how I was introduced to the concept (I learned about abstract vector spaces and transformations and how they are represented by matrices - and only then about determinants and their properties and different ways of computing them). That's why Axler's critique rings somewhat hollow to me - maybe it's different if you learn about linear algebra mostly in an engineering context as opposed to pure maths.
For theoretical purposes, determinants are important because they constitute a homomorphism between the general linear group and the multiplicative group of the underlying field, and the kernel of this homomorphism is the special linear group. This leads to some very short and elegant proofs.