Take the 2-minute tour ×  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.