# Determinant applications for 16 year olds

I am teaching matrices, determinants and systems to a class of 16-year-olds.

As you can expect, they do not know calculus or linear algebra.

What they do know is a little bit of matrices (how to multiply and invert them, for instance), some plane geometry, some spacial geometry, some analytical geometry, and most elementary functions: linear, quadratic, exponential, modular, trigonometric.

I had to explain what subjects they have studied because it must vary wildly from country to country.

In this context, I am having great trouble in finding real-world applications and examples of determinants being applied to do anything at all. The only examples I could show are:

Area of a triangle given the coordinates A,B,C; Cramer's rule for solving systems of linear equations. That being said, if anyone can think of any example I could use in my class, please tell me! Thanks in advance for your help.

## 2 Answers

I think you can go deeper than area of a triangle. This leads to computing the area of a polygon, and the volume of a polyhedron, two very common and important calculations. Wolfram Demo by Ed Pegg
See also this discussion re determinants here on MESE. Nearly every video game uses determinants somewhere deep in the code: The signed tetrahedron volume indicates whether the apex is to the positive or negative side of the plane containing the base. And deciding sidedness is a key calculation for collision detection.

I'm not sure if you would count it as a 'real world' example, but you have enough to define the Alexander polynomial and so prove (modulo invariance) the non-triviality of small knots (or bigger ones with a computer).