Preface
Chapter 1
Chapter 2

Additional Material for this Chapter
Exercises for this Chapter
Answers to Exercises

Chapter 3
Determinants

From a square matrix a single number can be extracted. In the previous chapter the rank was such a number. In this chapter another number called a "determinant" will be extracted. The discussion is not extensive enough to be complete, but carries the idea of "a brief introduction."

A starting place is the evaluation of the determanent of a 2x2 matrix:
[1]

                                |a    b|       =     ad - cb  
                                |c    d|
Therefore
[2]
                                |3    4|       =       (3)(5) - (2)(4) = 15 - 8 =  7
                                |2    5|
and
[3] 
                               |10   4|       =       14
                               |9    5|
and
[4]
                              |3   10|        =       7
                              |2    9|
The numbers in these determinants [2],[3],[4] are taken from the equations
[5]
			3x + 4y   =  10
			2x + 5y   =   9

It is not coincidence that quotients of these determinants give solutions to equations [5]:
[6] Solving for x:
                             |10    4|   
                             | 9    5|     
                            ___________   =  14/7   =   2  =  x
                             |3     4|       
                             |2     5|
[7] Solving for y:
                            |3    10|
                            |2     9|
                           ___________    =  7/7   =   1   =   y
                            |3     4|       
                            |2     5|

There is a pattern here that makes it easier to remember the locations of the determinants. Temporarily colors are used here to make the patterns more discernable. The columns in the determinants are colored red coefficients of x), blue (coefficients of y) and black (right hand values of the equations).

The denominators are determinants of coefficients with the redand blue columns. To solve for x, replace the red column by the black column to get the numerator determinant.. To solve for y, replace the