Comment by RossBencina
14 hours ago
I don't think determinants play a central role in modern advanced matrix topics.
Luckily for me I read Axler's "Linear Algebra Done Right" (which uses determinant-free proofs) during my first linear algebra course, and didn't concern myself with determinants for a very long time.
Edit: Beyond cofactor expansion everyone should know of at least one quick method to write down determinants of 3x3 matrices. There is a nice survey in this paper:
Dardan Hajriza, "New Method to Compute the Determinant of a 3x3 Matrix," International Journal of Algebra, Vol. 3, 2009, no. 5, 211 - 219. https://www.m-hikari.com/ija/ija-password-2009/ija-password5...
The 4th edition of Linear Algebra Done Right has a much improved approach to determinants themselves (still relegated to the end, where it should be). From the list of improvements:
> New Chapter 9 on multilinear algebra, including bilinear forms, quadratic forms, multilinear forms, and tensor products. Determinants now are defned using a basis-free approach via alternating multilinear forms.
The basis-free definition is really rather lovely.
> I don't think determinants play a central role in modern advanced matrix topics.
Not true at all. It's integral to determinantal stochastic point processes, commute distances in graphs, conductance in resistor networks, computing correlation via linear response theory, enumerating subgraphs, representation theory of groups, spectral graph theory... I am sure many more