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?
One great geometric significance of the determinant is that it helps keep track of how the area of a region is scaled under the linear transformation. Here is a website that allows you to apply different linear transformation: http://math.mercyhurst.edu/~lwilliams/Applets/algebra/linearTransformations.php. Look at how the determinant changes and also the area. The determinant is the amount the area gets scaled under the linear transformation.