Fundamentals of Matrix Algebra, 2011a

3.3 The Determinant
. Example 68 Find the determinant of A, B and C where
A =
[ ] 1 2
, B =
3 4
[ ] 3 −1
and C =
2 7
[ ] 1 −3
.
−2 6
S Finding the determinant of A:
det (A) =
∣ 1 2
3 4 ∣
= 1(4) − 2(3)
= −2.
Similar computaons show that det (B) = 3(7) − (−1)(2) = 23 and det (C) =
1(6) − (−3)(−2) = 0. .
Finding the determinant of a 2 × 2 matrix is prey straighorward. It is natural to
ask next “How do we compute the determinant of matrices that are not 2 × 2?” We
first need to define some terms. 15
.
Ḋefinion 24
Matrix Minor, Cofactor
Let A be an n × n matrix. The i, j minor of A, denoted A i,j , is
the determinant of the (n − 1) × (n − 1) matrix formed by
deleng the i th row and j th column of A.
.
The i, j-cofactor of A is the number
C ij = (−1) i+j A i,j .
Noce that this definion makes reference to taking the determinant of a matrix, while
we haven’t yet defined what the determinant is beyond 2 × 2 matrices. We recognize
this problem, and we’ll see how far we can go before it becomes an issue.
Examples will help.
. Example 69 .Let
⎡
⎤
⎡
A = ⎣ 1 2 3 ⎤
1 2 0 8
4 5 6 ⎦ and B = ⎢ −3 5 7 2
⎥
⎣ −1 9 −4 6 ⎦ .
7 8 9
1 1 1 1
Find A 1,3 , A 3,2 , B 2,1 , B 4,3 and their respecve cofactors.
15 This is the standard definion of these two terms, although slight variaons exist.
137