Determinants as proportions

This is a continuation of a discussion of determinants started here. And there is more here.

Proportions

So if determinants are important how important are they and when can they be introduced reasonably. One place that they could possibly come up is in reference to proportions. As a reminder, a proportion is equation where two ratios are equal. One method to solve equations such as this is the Means-Extremes Property. This is more commonly known as “Cross-Multiplying”. I think I speak for many teachers that cross-multiplying is a “bane of existence”. Cross-multiplying is a rule that is often overused. It seems to quickly rise to the top of all students list of favorite methods so that whenever in doubt about how to proceed in a problem with fractions teachers often here the idea put forth that the correct method might include cross-multiplying. This is probably because few students really understand what this method does or why it works

Connections between determinants and proportions

So what is the connection between determinants and cross multiplying. Well
it can be seen from a variety of ways. First is in the formulas themselves. A
proportion has the form:

after cross-multiplying we know that

this last equation can be rewritten in terms of determinants as

In a proportion we are given that two fractions are the same. Each of those fractions can be thought of as vectors. Similar to the definition of slope as a fraction or as a vector. With this definition of the fraction we see that the two fractions will be equivalent if their vectors point in the same direction. If they point in the same direction then the area given by the determinant will be zero.

Advantages of the Determinant Formulation

The advantage of this determinant method to solving proportions is that we eliminate the fractions from the problem. Cross-multiplying would only exist in determinants, where the rightly do play a role. Students would be less likely to misapply the idea of cross-multiplication to every situation with fractions. Determinants would be introduced earlier and their presentation of area would be well supported. Students would also have to have a clearer understanding of slope as a primary way of looking at fractions.

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)