Page 24 - Vector Analysis

20 CHAPTER 1. Linear Algebra

Definition 1.67 (Minor, Cofactor, and Adjoint matrices). Let A be an n ˆ n matrix, and
A(ˆi, ˆj) be the (n ´ 1) ˆ (n ´ 1) matrix obtained by eliminating the i-th row and j-th column

of A; that is,


a11 a12 ¨ ¨ ¨ a1(j´1) a1(j+1) ¨ ¨ ¨ a1n

A(ˆi, ˆj) =  ... ... ¨¨¨ ... ... ¨¨¨ ...  .
¨¨¨
a(i´1)1 a(i´1)2 a(i´1)(j´1) a(i´1)(j+1) ¨¨¨ a(i´1)n
a(i+1)1 a(i+1)2 a(i+1)(j+1) ... a(i+1)n
a(i+1)(j´1)
... ... ... ... ...

an1 an2 ¨ ¨ ¨ an(j´1) an(j+1) ¨ ¨ ¨ ann

The (i, j)-th minor of A is the determinant of A(ˆi, ˆj), and the (i, j)-th cofactor, is the

(i, j)-th minor of A multiplied by (´1)i+j. The adjoint matrix of A, denoted by Adj(A),

is the transpose of the cofactor matrix; that is,

[] = (´1)i+j det (A(ˆj,ˆi)) .
Adj(A) ij
  
12 3 ´3 ´3 6
´1 2 ,
Example 1.68. Let A =  3 ´1 2 . Then the minor matrix of A is ´8

0 2 ´1 7 ´7 ´7
 
´3 3 6 ´3 8 7

the cofactor matrix of A is  8 ´1 ´2, and the adjoint matrix of A is  3 ´1 7 .

7 7 ´7 6 ´2 ´7

The following lemma provides a way of computing the minors of a matrix.

Lemma 1.69. Let A be an n ˆ n matrix. Then

det (A(ˆi, ˆj)) = (´1)i+j ÿ ź

εk1k2¨¨¨kn aℓkℓ .

(k1,¨¨¨ ,kn)PP(n), ki=j 1ďℓďn
ℓ‰i

Proof. Fix (i, j) P t1, 2, ¨ ¨ ¨ , nu ˆ t1, 2, ¨ ¨ ¨ , nu. The matrix A(ˆi, ˆj) is given by A(ˆi, ˆj) = [bαβ],
where α, β = 1, 2, ¨ ¨ ¨ , n ´ 1, and

$ aαβ if α ă i and β ă j,
’ if α ą i and β ă j,
if α ă i and β ą j,
’ if α ą i and β ą j.

’ a(α+1)β
’
&

bαβ = aα(β+1)

’
’

’

’

% a(α+1)(β+1)

Each permutation (σ1, σ2, ¨ ¨ ¨ , σn´1) of degree n ´ 1 corresponds a unique permutation
(k1, k2, ¨ ¨ ¨ , kn) of degree n such that