Determinant applications for 16 year olds

Question

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.

Answer 1

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.

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      ^{Wolfram Demo by Ed Pegg}

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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.

Answer 2

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).