Determinant Demonstraition

[bergerud]

Ok, math lesson time! The determinant of a 2 X 2 matrix (ad - bc) gives the area of the parallelogram spanned by the two 2D vectors (a,b) and (c,d) which are the rows of the matrix. This is proved with algebra and the students have no idea why it is geometrically true. For years, I have told my students there must be a way draw triangles and see that it is true. I had never done it but I new it must be possible because it is true! Well I finally put some time into it and discovered a most pleasing geometric proof. I have never, in my 25 years of teaching seen this proof before. I is so simple, I cannot understand why it is not all over the internet.

So here is my demo carved out of walnut with arbutus triangles. Can you see why the area of the parallelogram is equal to the determinant (difference of the area of the large square and the small square) ?