5.1. Introduction

Determinants of order 2 and 3 were defined in Chapter 2 by the formulae

        |    a 11 a 12 a 21 a 22    |= a 11 a 22 a 12 a 21                       (5.1)

        |    a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33    |= a 11 |    a 22 a 23 a 32 a 33    | a 12 |    a 21 a 23 a 31 a 33    |+ a 13 |    a 21 a 22 a 31 a 32    |               (5.2)

The similarity in form between determinants and matrices suggests a close relationship between them.  Indeed, we shall treat the determinant as a function that maps a square matrix to a number, i.e.,

        det: M n,n R  or  C           by  AdetA=|    a ik    |

where M n,n  is the space of all n ´ n matrices and the function value detA=|    a ik    |  is to be evaluated by some generalized version of eqs(5.1-2).

 

There are many different, but equivalent, ways to carry out the generalization.  One way that emphasizes the anti-symmetry of the determinant function is to define it as

        detA=det ( a ik ) i,k=1 n,n   =|    a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn    |   = P ( ) P a 1 P 1 a 2 P 2 a n P n

where P is a permutation of n integers, i.e., P( 1,2,,n )=( P 1 , P 2 ,, P n )  and

( ) P ={ +1 1    if P is an even odd  permutation.

 

In general, there are n! terms in the sum.  For example, the case n=3  has 6 terms:

       

P1

P2

P3

(-)P

1

2

3

+1

1

3

2

-1

2

1

3

-1

2

3

1

+1

3

1

2

+1

3

2

1

-1

so that

        det ( a ik ) i,k=1 3,3 = a 11 a 22 a 33 a 11 a 23 a 32 a 11 a 21 a 33 + a 11 a 23 a 31 + a 11 a 21 a 32 a 11 a 22 a 31

as can be checked against (5.2) with the help of (5.1).

 

On the other hand, there is also a geometrical interpretation of a determinant as a volume in E n .  This will play significant roles in many area of mathematics, e.g., multi- dimensional integrals, and exterior derivatives.  In the following sections, we shall explore some elementary features of this aspect of the determinant function.