Page 276 - Advanced engineering mathematics

256 CHAPTER 8 Determinants

This method, called expansion by cofactors, can be used recursively until we have determinants
of small enough size to be easily evaluated.
Choose a row k of A =[a ij ]. An extension of property (7) of determinants from Section 8.1
enables us to write

a 11 ··· ··· a 11 ··· ···
a 12 a 1n a 12 a 1n
. . . . . . . .

. . . . . . . . . . . . . . . . . .
.
.
.
.

|A|=|[a ij ]| = a k1 a k2 ··· ··· a kn = a k1 0 ··· ··· 0

. . . . . . . .

. . . . . . . . . . . . . . . . . .
.
.
.
.

a n1 a n2 ··· ··· a nn a n1 a n2 ··· ··· a nn

··· ··· ··· ···
a 11 a 11
a 12 a 1n a 12 a 1n

. . . . . . . .
. . . . . . . . . . . . . . . . . .
.
.
.
.

+ 0 a k2 ··· ··· 0 + ··· + 0 0 ··· ··· a kn .

. . . . . . . .
. . . . . . . . . . . . . . . . . .
.
.
.
.

a n1 a n2 ··· ··· a nn a n1 a n2 ··· ··· a nn
Each of the n determinants on the right has a row with exactly one possibly nonzero element,
and can be expanded by that element, as in Section 8.2. To write this expansion, define the minor
of a ij to be the determinant of the n − 1 × n − 1 matrix formed by deleting row i and column j
of A. This minor is denoted M ij .The cofactor of a ij is the number (−1) i+ j M ij . Now this sum of
determinants gives us the following theorem.
THEOREM 8.2 Cofactor Expansion by a Row
For any k with 1 ≤ i ≤ n.
n
k+ j
|A|= (−1) a kj M kj . (8.4)
j=1
Equation (8.4) states that the determinant of A is the sum, along any row k, of the matrix ele-
ments of that row, each multiplied by its cofactor. This holds for any row of the matrix, although
of course this sum is easier to evaluate if we choose a row with as many zero elements as possi-
ble. Equation (8.4) is called expansion by cofactors along row k. If we write out a few terms for
fixed k we get
|A|= (−1) k+1 a k1 M k1 + (−1) k+2 a k2 M k2 + ··· + (−1) k+n a kn M kn .
EXAMPLE 8.4
Let
⎛ ⎞
−6 3 7
A = ⎝ 12 −5 −6 ⎠
2 4 −6
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October 14, 2010 14:26 THM/NEIL Page-256 27410_08_ch08_p247-266