Very good question, I think. There is indeed a backlash against determinants, as evidenced for instance by Sheldon Axler's book Linear Algebra Done Right. To quote Axler from this link,

The novel approach used throughout the book takes great care to motivate concepts and simplify proofs. For example, the book presents, without having defined determinants, a clean proof that every linear operator on a finite-dimensional complex vector space (or on an odd-dimensional real vector space) has an eigenvalue.

Indeed. If you think that determinants are taught in order to invert matrices and compute eigenvalues, it becomes clear very soon that Gaussian elimination outperforms determinants in all but the smallest instances.

On the other hand, my undergraduate education spent a tremendous amount of time on determinants (or maybe it just *felt* that way). We built the theory (from $\dim \Lambda^n(K^n)=1$), proved some interesting theorems (row/column expansion, bloc-determinants, derivative of det(A(x))), but never used determinants to perform prosaic tasks. Instead, we spent a lot of time computing $n \times n$ determinants such as Vandermonde (regular and lacunary), Cauchy, circulant, Sylveser (for resultants)... and of course, giving a few different proofs of the Cayley-Hamilton theorem!

What's the moral of the story? I think it's twofold:

Determinants are mostly a concern from a theoretical point of view. Computationally, the definition is awful. The most sensible thing to do to compute a determinant is to use Gaussian elimination, but if you're going to go through that bother, chances are that it's not really the determinant that you're after but rather something else that elimination will give you.

Determinants are a fertile ground to get to grips with a lot of really fundamental mathematical tools that a student of abstract mathematics should know backwards and forward. If you do everything I described above, you must learn deep results about the structure of the symmetric group $S_n$ (and more generally about multilinear forms), 10 flavors of induction, practical uses of group morphisms (from $GL_n(K)$ to $K^\star$). And of course, the existence of determinants itself is crucial to the more theoretical developments such a student will encounter later on.

I've had pure math undergrads who had learned linear algebra from Axler's book. They knew how to compute a determinant. They had no idea why anyone would want to. So determinants are **still** a big deal, but just for the right audience: I'm perfectly fine with most scientist and engineers ignoring determinants beyond $3\times 3$. Mathematics students, especially those with a theoretical bent, can learn a lot from determinants.

`$$\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}, $$`

it's little consolation that the minimal polynomial exists by an abstract argument! To me, this doctrinal approach appears just as fruitless as the attempts to base real analysis on "constructible" numbers only (since countably many reals expressible in a finite way "should be sufficient"). – Victor Protsak Aug 18 '10 at 20:24