Page 238 - Linear Algebra Done Right

Determinant of a Matrix
vice versa, for a net change (so far) of 1 or −1 (both odd numbers) in
the number of pairs not in their natural order.
Consider each entry between the two interchanged entries. If an in- 229
termediate entry was originally in the natural order with respect to the
first interchanged entry, then it no longer is, and vice versa. Similarly,
if an intermediate entry was originally in the natural order with respect
to the second interchanged entry, then it no longer is, and vice versa.
Thus the net change for each intermediate entry in the number of pairs
not in their natural order is 2, 0, or −2 (all even numbers).
For all the other entries, there is no change in the number of pairs
not in their natural order. Thus the total net change in the number of
pairs not in their natural order is an odd number. Thus the sign of the
second permutation equals −1 times the sign of the first permutation.

If A is an n-by-n matrix
 
a 1,1 ... a 1,n
 . . 
10.24 A =  . . . .  ,
 
a n,1 ... a n,n
then the determinant of A, denoted det A, is defined by Our motivation for this

definition comes
10.25 det A = sign(m 1 ,...,m n ) a m 1 ,1 ...a m n ,n .
from 10.22.
(m 1 ,...,m n )∈perm n
For example, if A is the 1-by-1 matrix [a 1,1 ], then det A = a 1,1 be-
cause perm 1 has only one element, namely, (1), which has sign 1. For
a more interesting example, consider a typical 2-by-2 matrix. Clearly
perm 2 has only two elements, namely, (1, 2), which has sign 1, and
(2, 1), which has sign −1. Thus

a 1,1 a 1,2
10.26 det = a 1,1 a 2,2 − a 2,1 a 1,2 .
a 2,1 a 2,2
To make sure you understand this process, you should now find the
formula for the determinant of the 3-by-3 matrix The set perm 3
contains 6 elements. In
 
a 1,1 a 1,2 a 1,3
general, perm n
 
 a 2,1 a 2,2 a 2,3 
contains n! elements.
a 3,1 a 3,2 a 3,3
Note that n! rapidly
using just the definition given above (do this even if you already know grows large as n
the answer). increases.