Math21b, Spring 2007, Linear Algebra and Differential Equations,

A bit more than 10 years ago, an article Sheldon Axler, "Down with determinants" in the American Mathematical Monthly 102 (1995), 139-154, challenged to teach determinants in linear algebra courses. Read it here.

While Axlers article is interesting and important because any attempt to shake up dogmas in teaching or break up rusty old structures should be welcomed, the proposed alternative did not catch on. Here are some reasons, why determinants are needed.

They give explicit formulas for the inverse of a matrix or the solution x of Ax=b. These are not effective methods to compute the inverse or the solution but if we want to do an error analysis with parameters, then explicit formulas are great.
Determinants are very natural objects. Every real valued function on square matrices which satisfies f(AB)=f(A) f(B), f(1)=1, f(k A) = kⁿ A is the determinant.
Determinants are useful also in multilinear algebra and differential geometry.
Determinants allow to define an orientation of a basis, if det(A) is positive, then the basis has a positive orientation, if det(A) is negative, then the orientation is negative. They have a geometric meaning of a signed volume.

Determinants provide a clean way to define the characteristic polynomial det(x 1 - A) =0. It gives an immediate link between the eigenvalue and the existence of a kernel of A-x 1.
Determinants are important in many aspects of physics. This reason alone would make an omission foolish.
The computation of determinants is one of the most fun topics in linear algebra. Finding the best way to crack a determinant has aspects of puzzles. Cracking a large determinant can be as fun as solving a crossword puzzle or Sudoku.
Determinants are a fantastic tool to teach algorithms and the effectiveness of the method. While the recursive definiton of the determinant needs n! steps, the best method is a polyonomial. Determinants give lesson in complexity theory. Nobody knows whether one can compute permanents effectively, which are determinants where the signs in the recursive Laplace definition are not alternating +,-,+, but everywhere +.