Determinants and Their Applications in Mathematical Physics

12 2. A Summary of Basic Determinant Theory

Similarly, the column operations

C

′

i=

i

∑

j=1

vijCj,vii=1, 1 ≤i≤ 3 ,vij=0, i>j, (2.3.6)

when performed on A 3 in reverse order, have the same effect as

postmultiplication ofA 3 byV

T

3

2.3.3 First Minors and Cofactors; Row and Column

Expansions

To each elementaijin the determinantA=|aij|n, there is associated a

subdeterminant of order (n−1) which is obtained fromAby deleting row

iand columnj. This subdeterminant is known as a first minor ofAand

is denoted byMij. The first cofactorAijis then defined as a signed first

minor:

Aij=(−1)

i+j

Mij. (2.3.7)

It is customary to omit the adjectivefirstand to refer simply to minors and

cofactors and it is convenient to regardMijandAijas quantities which

belong toaijin order to give meaning to the phrase “an element and its

cofactor.”

The expansion ofAby elements from rowiand their cofactors is

A=

n

∑

j=1

aijAij, 1 ≤i≤n. (2.3.8)

The expansion of Aby elements from columnj and their cofactors is

obtained by summing overiinstead ofj:

A=

n

∑

i=1

aijAij, 1 ≤j≤n. (2.3.9)

SinceAijbelongs to but is independent ofaij, an alternative definition of

Aijis

Aij=

∂A

∂aij

. (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

∑

j=1

aijAkj=0, 1 ≤i≤n, 1 ≤k≤n, k=i. (2.3.11)