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8.3 Evaluation of Determinants II 257

If we expand by cofactors along row one, we get

3

|A|= (−1) 1+ j a 1 j M 1 j
j=1

−5 −9 12 −9

=(−1) 1+1 (−6) + (−1) 1+2 (3)
4 −6 2 −6

12 −5

+ (−1) 1+3 (7)
2 4

=(−6)(30 + 36) − 3(−72 + 18) + 7(−48 + 10) = 172.
If we expand by row three, we get
3

|A|= (−1) 3+ j a 3 j M 3 j
j=1

3 7 −6 7

=(−1) 3+1 (2) + (−1) 3+2 (4)
−5 −9 12 −9

−6 3

+ (−1) 3+3 (−6)
12 −5

=(2)(−27 + 35) − 4(54 − 84) − 6(30 − 36) = 172.
We can also do a cofactor expansion along a column. Now fix j and sum the elements of
column j times their cofactors.

THEOREM 8.3 Cofactor Expansion by a Column

For any j with 1 ≤ j ≤ n,

n

|A|= (−1) i+ j a ij M ij . (8.5)
i=1

EXAMPLE 8.5
We will expand the determinant of the matrix of Example 8.3, using column 1:

3
i+1
|A|= (−1) a i1 M i1
i=1

−5 −9 3 7
=(−1) 1+1 (−6) + (−1) 2+1 (12)
4 −6 4 −6

3 7
3+1

+ (−1) (2)
−5 −9
=(−6)(30 + 36) − 12(−18 − 28) + 2(−27 + 35) = 172.

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October 14, 2010 14:26 THM/NEIL Page-257 27410_08_ch08_p247-266