Page 80 - A Course in Linear Algebra with Applications

64 Chapter Three: Determinants

and indeed

2
P 3
w

Definition of a determinant in general
We are now in a position to define the general n x n
determinant. Let A = (ay) n ,n be an n x n matrix over some
field of scalars. Then the determinant of A is the scalar defined
by the equation

det(A) = Y^ s ign(i 1,i2,...,i n)a Ula 2i 2

where the sum is taken over all permutations ii,%2, • • • ,i n of
1,2,...,n.
Thus det(.A) is a sum of n\ terms each of which involves
a product of n elements of A, one from each row and one
from each column. A term has a positive or negative sign
according to whether the corresponding permutation is even or
odd respectively. One determinant which can be immediately
evaluated from the definition is that of I n:

det(In) = 1.

This is because only the permutation 1,2,... ,n contributes a
non-zero term to the sum that defines det(J n ).
If we specialise the above definition to the cases n =
1,2,3, we obtain the expressions for det(^4) given at the be-
ginning of the section. For example, let n — 3; the even and
odd permutations are listed above in Example 3.1.2. If we
write down the terms of the determinant in the same order,
we obtain