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 parallelpiped.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?