TITLE: Why were matrix determinants once such a big deal?
QUESTION [107 upvotes]: I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. I am curious about why determinants were such a big deal. So here's my question:
a) What are examples of cool tricks you can use matrix determinants for? (Cramer's rule comes to mind, but I can't come up with much more.) What kind of amazing properties do matrix determinants have that made them such popular objects of study?
b) Why did the use of matrix determinants fall out of favor? Some back history would be very welcome.
Update: From the responses below, it seems appropriate to turn this question into a community wiki. I think it would be useful to generalize the original series of questions with:
c) What significance do matrix determinants have for other branches of mathematics? (For example, the geometric significance of the determinant as the signed volume of a parallelepiped.) What developments in mathematics have been inspired by/aided by the theory of matrix determinants?
d) For computational and theoretical applications that matrix determinants are no longer widely used for today, what have supplanted them?

REPLY [6 votes]: The area of an arbitrary triangle is most easily computed from its vertex coordinates as a 3x3 determinant (no square roots, trig, etc.).  This generalizes to 3D (volume  of a tetrahedron) and higher dimensions.
The cross product of two vectors a,b (essential tool in mechanics) is easily understood and memorized once one observes that its coordinates are 2x2 minor determinants of a 2x3 array formed by a and b.  This too generalizes and provides a definition for cross product of n-1 vectors in n dimensions.
More generally, determinants combined with homogeneous coordinates are all one needs to derive elegant formulas for the basic operations of n-dimensional projective geometry, such as plane through 3 points, intersection of 3 planes.  They also provide a coordinate representation (Pluecker coordinates) for lines in 3-space, or more generally for k-dim subspaces of n-dim prijective space.
The sign of the determinant of a matrix tells whether the rows are a left- or right-handed frame, and whether the corresponding linear map preserves or reverses orientations.
Determinants are more efficient than Gaussian elimination to compute the inverse of a 2x2 or 3x3 matrix, perhaps even 4x4. Unlike standard G.E. these formulas do not require division and may require fewer bits per number in intermediate results.

REPLY [3 votes]: I quickly scanned the answers and didn't find the following:
Square matrices of order $n$ can be considered to be embedded in $\Bbb R^{n \times n}$. The determinant is a continuous function of the entries of the matrix, so the singular matrices are $\det^{-1}(0)$ and are therefore a closed set in $\Bbb R^{n \times n}$. Thus, there will be non-singular matrices arbitrarily close to a singular matrix in any convenient metric on $\Bbb R^{n \times n}$. Of course, they will have pretty ugly condition numbers.