Determinants

Determinants are interesting part of mathematics. They are an important measure of a transformation, they form a way to compute the cross product, my mathematical training says that they are connected with character theory. But how do I help student's understand their significance. Determinants have not been around very long, getting started about 1750. If I don't know much about matrices, transformations, or characters, then what are determinants. Until we can answer that question I don't know how to teach determinants. I don't want to just teach an algorithm to compute a number. We must have reasons for it.

Here is a reason that I found while reading mathworld. It is possible to create determinant equations that immediately give equations for lines, circles, parabolas given the right number of points. This emphasizes several of the important properties of determinants: linearity, and the property that if two rows are identical then the determinant has value zero. In fact, it is this last fact that is the key point of this. Let's see some examples. We start with equation for a line. (Bear with me, I don't know how to format the math yet.)

If I want to find the equation of a line through the points (3, 2) and (5, 6) then I can set up the following determinant:

| x  y  1 |
| 3  2  1 | = 0
| 5  6  1 |

First, this is a linear equation by the linearity of the determinant, and if we substitute x=3 and y=2 into this determinant then we get a true equation because of the property that if two rows are identical then the determinant is zero, similarly with (5, 6).

When you expand the determinant you get -4x+2y+8=0. This is not quite standard form, but close enough.

What is more amazing is that this trick works for other types of equations such as parabolas. In the case of the parabola you use a determinant of the form:

| x^2 x y 1 |
| a^2 a b 1 |
| c^2 c d 1 | = 0
| e^2 e f 1 |

where (a, b), (c, d), and (e, f) are points on the parabola.

There is even a version for a circle given three points and on mathworld there is the general case for any conic given 5 points.

I like the way that this approach unifies these different processes and emphasizes the important qualities of the determinant, but I am still left with my big question. What is a determinant? (Clear and Concise please)