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.