College Algebra 9th txtbk

SECTION 7–5 Determinants and Cramer’s Rule 489
Don’t panic! You don’t need to memorize formula (2). After we introduce the ideas of
minor and cofactor below, we will state a theorem that can be used to obtain the same result
with much less trouble.
The minor of an element in a third-order determinant is a second-order determinant
obtained by deleting the row and column that contains the element. For example, in the
determinant in formula (2),
Deletions are usually done mentally.
Minor of a 23
a 11 a 12
` `
a 32
a 31
a 11 a 12 a 13
† a 21 a 22 a 23 †
a 31 a 32 a 33
a 11 a 32 a 31 a 12
a 11 a 12 a
a 13
11 a 13
Minor of a 32 ` ` † a 21 a 22 a 23 † a 11 a 23 a 21 a
a 13
21 a 23
a 31 a 32 a 33
ZZZ EXPLORE-DISCUSS 1
Write the minors of the other seven elements in the determinant in formula (2).
A quantity closely associated with the minor of an element is the cofactor of an element
a ij (from the ith row and jth column), which is defined as the product of the minor
of a and (1) ij ij .
Z DEFINITION 1 Cofactor
Cofactor of a ij (1) ij (Minor of a ij )
So a cofactor is just a minor with either a positive or negative sign. The sign is determined
by raising 1 to a power that is the sum of the numbers indicating the row and column
in which the element appears. Note that (1) ij is 1 if i j is even and 1 if i j
is odd. So if we are given the determinant
then
a 11 a 12 a 13
† a 21 a 22 a 23 †
a 31 a 32 a 33
Cofactor of a 23 (1) 23 a 11 a 12 a 11 a 12
` ` ` ` (a 11 a 32 a 31 a 12 )
a 31 a 32 a 31 a 32
Cofactor of a 11 (1) 11 a 22 a 23 a 22 a 23
` ` ` ` a 22 a 33 a 32 a 23
a 33 a 32 a 33
a 32
EXAMPLE 2 Finding Cofactors
Find the cofactors of 2 and 5 in the determinant
2 0 3
† 1 6 5 †
1 2 0