Determinant of a Matrix

Date: 11/05/97 at 20:38:18
From: Mark Geraghty
Subject: Definition of DETERMINANT

My Alegebra 2 teacher told us that for extra credit we could give him 
a complete and unabridged version of the definition of a DETERMINANT 
of a matrix. He said that we could only find it in a library or an 
advanced collegiate math book. He also said that if we could 
understand the definition, it wasn't the right one. 

I have tried numerous dictionaries and other references over the 
Internet. Please help me if you can! Thank you so much.

Date: 11/06/97 at 14:30:31
From: Doctor Tom
Subject: Re: Definition of DETERMINANT

Hi Mark,

I'll give you two definitions that are exactly equivalent, but sound
very different.

The first is geometric. I assume you've plotted things in an x-y
coordinate system, right? I assume you can imagine doing the same 
thing in three dimensions with an x-y-z coordinate system as well.

In 2-D, when you talk about the point (2, 4), you can think of the
"2" and "4" as directions to get from the origin to the point -
"move 2 units in the x direction and 4 in the y direction."  In
a 3-D system, the same idea holds - (1, 3, 7) means start at the
origin (0,0,0), go 1 unit in the x direction, 3 in the y direction,
and 7 in the z direction.

Similarly, you could have coordinates in one dimension, but there's 
just one number.

The determinant of a 1x1 matrix is the signed length of the line from 
the origin to the point. It's positive if the point is in the positive
x direction, negative if in the other direction.

In 2-D, look at the matrix as two 2-dimensional points on the plane,
and complete the parallelogram that includes those two points and the 
origin. The (signed) area of this parallelogram is the determinant.  
If you sweep clockwise from the first to the second, the determinant 
is negative; otherwise, positive.

In 3-D, look at the matrix as 3 3-dimensional points in space.  
Complete the parallepiped that includes these points and the origin, 
and the determinant is the (signed) volume of the parallelepiped.

The same idea works in any number of dimensions.  The determinant
is just the (signed) volume of the n-dimensional parallelepiped.

Notice that length, area, volume are the "volumes" in 1-, 2-, and 
3-dimensional spaces.  A similar concept of volume exists for 
Euclidean space of any dimensionality.

Okay. That's the geometric definition. I like it because I can make a 
mental picture of it. Here's the algebraic definition:

I'll do it in 3 dimensions, but exactly the same idea works in any
number of dimensions.  Let's look at the determinant of this matrix:

  | a11 a12 a13 |
  | a21 a22 a23 |
  | a31 a32 a33 |

The numbers after the "a" are the row and column numbers.

A permutation of a set of numbers is a re-arrangement.

For example, there are 6 permutations of the list (1 2 3), including
the "re-arrangement" that leaves everything unchanged). Ignore for the 
moment the "+1" and "-1" after each one:

(1 2 3) -> (1 2 3)   +1
(1 2 3) -> (1 3 2)   -1
(1 2 3) -> (2 1 3)   -1
(1 2 3) -> (2 3 1)   +1
(1 2 3) -> (3 1 2)   +1
(1 2 3) -> (3 2 1)   -1

Now imagine that you start with three objects labelled 1, 2, and 3
arranged as they are on the left, and need to convert them to the
order on the right, but you're only allowed to swap one pair at a 
time. To get to the final arrangement, you'll find that there are lots 
of ways to do it, but every way (for a particular rearrangement) 
always requires an even number of swaps or always requires an odd 
number of swaps. I've labelled those that always need an even number 
of swaps with +1 and those needing an odd number as -1 above.

Now write down 6 products of the "a" terms, where the first number
for each term is 1, 2, 3 and the second number is the rearrangement
above for each of the six rearrangements.

Here's what they are, in the same order as above.  Be sure you
understand this step:

a11*a22*a33
a11*a23*a32
a12*a21*a33
a12*a23*a31
a13*a21*a32
a13*a22*a31

The determinant is just the sum of all 6 terms, but put a "+" in
front if the rearrangement is even, and a "-" in front if the
rearrangement required an odd number of swaps.

Here's the answer:

+a11*a22*a33 -a11*a23*a32 -a12*a21*a33
+a12*a23*a31 +a13*a21*a32 -a13*a22*a31

For a 4x4 matrix, there will be 24 rearrangments, like this:

(1 2 3 4) -> (3 2 4 1) +1
...

so there will be 24 terms in the expression of the determinant.

For a 5x5 matrix there are 120 rearrangements, so there will be 120 
terms in the determinant, and so on.

For an NxN matrix, there will be N! (N factorial) terms, where
factorial means you multiply together all the terms from N down to 1.  
For example, 5! = "5 factorial" = 5x4x3x2x1 = 120.

I hope this helps.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/