Why Does This Work? Qs about Determinants

Standard

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.

image

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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4 responses »

  1. 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.

  2. 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

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