Use of determinants I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants.  In the math textbooks I have, they simply show how to compute a determinant and some properties about them (i.e. Cramer's Rule), but not why they are used and how they work.
So my question is, how do they work and why/when would I know to use them to help solve something?

Here's another example of use of determinants:
Let $F$ be a field, let $K$ be a field containing $F$, and finite-dimensional as a vector space over $F$. Let $\alpha$ be an element of $K$. The map $T:K\to K$ given by $T(x)=\alpha x$ is a linear transformation. Given a basis for $K$ as a vector space over $F$, one can find a matrix $A$ representing $T$. The matrix depends on the basis chosen, but its determinant does not; it only depends on $\alpha$, and it's called the norm of $\alpha$ (strictly speaking, the norm of $\alpha$ from $K$ to $F$). And the norm is a very important concept in Field Theory and Algebraic Number Theory.
