We have an Intro to Matrix Operations at the beginning of Algebra 2.

Here’s my unit organizer. Targets are based on Quality Core Standards.

I remember being shown this trick to use “determinants” to find area of polygons in a plane.

But why does it work? Sure it’s a nice quick trick, but why?

Also, we evaluate determinants and use them to find inverses…but what is the value really? What have I actually found when I do this process?

Maybe I once knew. I don’t know now. But I’d like to know why because my kids will ask. And I want to do better than in the past.

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Pam, I am no so familiar with this geometric application of determinants. But when I show kids Cramer’s Rule to solve systems, we can show why the “trick” works by looking at the equations ax+by=c and dx+ey=f. Follow the algebra and Cramer’s Rule emerges.

In your example, perhaps label the bottom point (a,b), then you also have (a+c, b) and (a, b+d). Use these points rather than the numbers and see if a rule emerges.

Thanks Bob!

I learned this in math team in high school — we called it the “shoelace theorem.” We end up proving it in my multivariable calculus class (http://www.artofproblemsolving.com/Wiki/index.php/Shoelace_Theorem).

However I recall that I came up with a proof in high school — before I took MV Calculus. I think I was only able to prove it for convex polygons… I don’t totally remember what that proof was (that was forever ago), but I’m 99% sure it used the fact that when you take the determinant of a 2×2 matrix, you end up with the area of a particular parallelogram defined by the matrix.

I don’t remember it being a “simple” proof. But maybe it was?

Always,

Sam

Thanks Sam!