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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!
```
Associated Topics:
College Coordinate Plane Geometry
College Linear Algebra

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