principle of multiplication by the number of distinct ways a thing can be done can be
extended to more than just two or three somethings.
Apermutation of a set of objects represents some arrangement of the objects.
The number of different permutations of n-objects is n!(read n-factorial). This is
because there are nchoices for the first position of the arrangement, (n−1) choices
for the second position of the arrangement, (n−3) choices for the third position of
the arrangement, etc, and these quantities are being multiplied.
A transposition is an interchanging of the positions of two objects within an
arrangement of the set of objects. In examining all possible permutations of the
integers (1 , 2 , 3 ,.. ., n )one finds these permutations can be divided into a group rep-
resenting an even number of transpositions and another group representing the odd
number of transpositions. For example, in going from (1234.. .)to (2134.. .)represents
1 transposition and going from (1234 ,.. .)to (2314.. .) would be two transpositions,
etc.
The determinant of a n×nsquare matrix A= (aij)is denoted by either of the
symbols det A or |A|. The determinant is a single number given by either of the
summations
det A=|A|=
∑
(−1)mai 1 aj 2 ak 3... an column expansion
det A=|A|=
∑
(−1)ma 1 ia 2 ja 3 k···an row expansion
The single number det A=|A| is the sum of all possible products in which there
appears one and only one element from each row (or column) multiplied by the
appropriate plus or minus sign. The sigma sign Σdenotes a sum over all n!permu-
tations of the numbers (1 , 2 , 3 ,.. ., n ) and the integers (i, j, k,.. ., )represent distinct
permutations of the numbers from the set (1 , 2 , 3 ,.. ., n ). The appropriate plus or mi-
nus sign is assigned to each product within the sum and is based upon whether the
permutation (i, j, k,... , )is either even (+1) or odd (-1). That is, m= +1 if (i, j, k,.. ., )
represents an even number of transpositions associated with the set (1 , 2 , 3 ,.. ., n )and
m=− 1 if (i, j, k,... , )represents an odd number of transpositions associated with
the set (1 , 2 , 3 ,.. ., n ).
Example 10-15. The matrix A=
(
a 11 a 12
a 21 a 22
)
has the determinant
|A|=
∣∣
∣∣a^11 a^12
a 21 a 22
∣∣
∣∣= + a 11 a 22 −a 12 a 21
