Page 27 - Determinants and Their Applications in Mathematical Physics

12 2. A Summary of Basic Determinant Theory

Similarly, the column operations
i

C = v ij C j , v ii =1, 1 ≤ i ≤ 3, v ij =0, i>j, (2.3.6)

i
j=1
when performed on A 3 in reverse order, have the same effect as
postmultiplication of A 3 by V .
T
3
2.3.3 First Minors and Cofactors; Row and Column
Expansions

To each element a ij in the determinant A = |a ij | n , there is associated a
subdeterminant of order (n − 1) which is obtained from A by deleting row
i and column j. This subdeterminant is known as a first minor of A and
is denoted by M ij . The first cofactor A ij is then defined as a signed first
minor:
A ij =(−1) i+j M ij . (2.3.7)
It is customary to omit the adjective first and to refer simply to minors and
cofactors and it is convenient to regard M ij and A ij as quantities which
belong to a ij in order to give meaning to the phrase “an element and its
cofactor.”
The expansion of A by elements from row i and their cofactors is

n

A = a ij A ij , 1 ≤ i ≤ n. (2.3.8)
j=1
The expansion of A by elements from column j and their cofactors is
obtained by summing over i instead of j:

n

A = a ij A ij , 1 ≤ j ≤ n. (2.3.9)
i=1
Since A ij belongs to but is independent of a ij , an alternative definition of
A ij is
∂A
A ij = ∂a ij . (2.3.10)

Partial derivatives of this type are applied in Section 4.5.2 on symmetric
Toeplitz determinants.

2.3.4 Alien Cofactors; The Sum Formula
The theorem on alien cofactors states that

n

a ij A kj =0, 1 ≤ i ≤ n, 1 ≤ k ≤ n, k = i. (2.3.11)
j=1