Linear Algebra, Theory And Applications, 2012a

86 DETERMINANTS
A permutation can also be considered as a function from the set
{1, 2, ··· ,n} to {1, 2, ··· ,n}
as follows. Let f (k) =i k . Permutations are of fundamental importance in certain areas
of math. For example, it was by considering permutations that Galois was able to give a
criterion for solution of polynomial equations by radicals, but this is a different direction
than what is being attempted here.
In what follows sgn will often be used rather than sgn n because the context supplies the
appropriate n.
3.3.2 The Definition Of The Determinant
Definition 3.3.4 Let f be a real valued function which has the set of ordered lists of numbers
from {1, ··· ,n} as its domain. Define
∑
f (k 1 ···k n )
(k 1 ,··· ,k n )
to be the sum of all the f (k 1 ···k n ) for all possible choices of ordered lists (k 1 , ··· ,k n ) of
numbers of {1, ··· ,n} . For example,
∑
f (k 1 ,k 2 )=f (1, 2) + f (2, 1) + f (1, 1) + f (2, 2) .
(k 1 ,k 2 )
Definition 3.3.5 Let (a ij )=A denote an n × n matrix. The determinant of A, denoted
by det (A) is defined by
det (A) ≡
∑
sgn (k 1 , ··· ,k n ) a 1k1 ···a nkn
(k 1,··· ,k n)
where the sum is taken over all ordered lists of numbers from {1, ··· ,n}. Note it suffices to
take the sum over only those ordered lists in which there are no repeats because if there are,
sgn (k 1 , ··· ,k n )=0and so that term contributes 0 to the sum.
Let A be an n × n matrix A =(a ij )andlet(r 1 , ··· ,r n ) denote an ordered list of n
numbers from {1, ··· ,n}. LetA (r 1 , ··· ,r n ) denote the matrix whose k th row is the r k row
of the matrix A. Thus
det (A (r 1 , ··· ,r n )) =
∑
sgn (k 1 , ··· ,k n ) a r1k 1
···a rnk n
(3.7)
and A (1, ··· ,n)=A.
(k 1 ,··· ,k n )
Proposition 3.3.6 Let (r 1 , ··· ,r n ) be an ordered list of numbers from {1, ··· ,n}. Then
sgn (r 1 , ··· ,r n )det(A) =
∑
(k 1 ,··· ,k n )
Proof: Let (1, ··· ,n)=(1, ··· ,r,···s, ··· ,n)sor