 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) = k^{n} 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 Ax 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 +.
