TITLE: What is the importance of determinants in linear algebra?
QUESTION [7 upvotes]: In some literature on linear algebra determinants play a critical role and are emphasized in the earlier chapters (see books by Anton & Rorres, and Lay). However in other literature it is totally ignored until the latter chapters (see Gilbert Strang).
How much importance should we give the topic of determinants? I tend to use it to find linear independence of vectors and might extend this to finding the inverse but I think Gauss Jordan and LU might be easier for inverse. Does it have any other uses in Linear Algebra.
Are there areas where determinants are used and have a real impact? Are there any real life applications of determinants?
Is there a really good motivating example or explanation which will hook students into this topic? 
In linear algebra, where should determinants be placed? Like I said in my comment - in some literature it is at the beginning whilst in others it is bolted on at the end. I like the idea of checking if vectors are independent by using determinants so think they should be placed before independence of vectors. What do you think? If you teach a linear algebra course where do you place this topic.

REPLY [5 votes]: This is quite an informal answer.
Determinants basically help to describe the nature of solutions of linear equations. The determinant of a real matrix is just some real number, telling you about the invertibility of the matrix and hence telling you things about linear equations wrapped up in the matrix.
The determinant being non-zero is equivalent to the matrix being invertible, which is equivalent to the corresponding sets of linear equations having EXACTLY one solution.
The determinant being zero means the matrix is NOT invertible. In this case the corresponding sets of linear equations can either have infinitely many solutions or none at all (depending on the numbers on the RHS).
So really the determinant is useful anywhere that linear equations crop up. For example, when checking linear independence, this is the same as demanding the existence of a UNIQUE solution to a set of linear equations (i.e. the zero vector solution). This is the same as the matrix determinant being non-zero as discussed above. Linear dependence must therefore be the same as the determinant being zero (so that there may be non-zero solutions to the equations, i.e. so that some of the vectors really can be made to add up non-trivially to give one of the others).