Finding Area Using a Determinant

Date: 05/25/2007 at 03:55:12
From: Richard
Subject: How does the determinant of a 2x2 matrix give the area?

I just can't see how this formula gives the area (determinant):

M = | 1 1|
    |-1 1|

det M = 1(1) - 1(-1) = 1 + 1 = 2, which is indeed the area of the
parallelogram defined by the two vectors (1,-1) and (1,1).

What are the relations in the formula?  I've tried putting them on
paper numerous times, but I just can't figure it out!

Date: 05/25/2007 at 17:10:45
From: Doctor Fenton
Subject: Re: How does the determinant of a 2x2 matrix give the area?

Hi Richard,

Thanks for writing to Dr. Math.  This isn't a trivial result.

This can be shown by some algebra and the dot product for vectors.  If
u = (x1,y1) and v = (x2,y2) are vectors, and ||u|| represents the
length of u, etc., then the dot product of u and v, u.v, is defined by

  u.v = x1*x2 + y1*y2 .

Using the Law of Cosines, it can be shown that

  u.v = ||u||*||v||*cos(@)

where @ is the angle between u and v.

The area of the parallelogram formed by u and v is given by


        ^---------------
       /|              /
   u  / |             /
     /  |h           /
    /   |           /
   /@   |          /
  ---------------->
          v
  |<--->|
     L 


  Area = h*||v||


since ||v|| is the length of the base, and h is the altitude as shown
in the diagram.

If L is the length of the projection of the vector u on the vector v
as shown, then

    L
  ----- = cos(@)  ,  or L = ||u||*cos(@) .
  ||u||

By Pythagoras, 

  h^2 = ||u||^2 - L^2   
      = ||u||^2 - (||u||*cos(@))^2

                                 (u.v)^2
      = ||u||^2 - ||u||^2 * -----------------
                             ||u||^2 * ||v||^2


Multiplying by ||v||^2 gives


  (Area)^2 = ||u||^2 * ||v||^2 - (u.v)^2

           = (x1^2 + y1^2)*(x2^2 + y2^2) - (x1*x2 + y1*y2)^2  


If you simplify this expression, you can show that this equation 
can be written as

  (Area)^2 = (x1*y2 - x2*y1)^2  ,

and the right side is the square of

   det [x1 y1]
       [x2 y2]   .

If you have any questions, please write back and I will try to explain
further.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/28/2007 at 04:14:37
From: Richard
Subject: Thank you (How does the determinant of a 2x2 matrix give the
area?)

Hey Dr. Fenton!  Thanks a million!  It all makes sense now!  I keep
forgetting to look at the geometry!