Determinants and Their Applications in Mathematical Physics

8 2. A Summary of Basic Determinant Theory

An=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

R 1

R 2

R 3

Rn

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

=

∣

∣C

1 C 2 C 3 ···Cn

∣

∣. (2.2.2)

The column vector notation is clearly more economical in space and will

be used exclusively in this and later chapters. However, many properties

of particular determinants can be proved by performing a sequence of row

and column operations and in these applications, the symbolsRiandCj

appear with equal frequency.

If every element inCjis multiplied by the scalark, the resulting vector

is denoted bykCj:

kCj=

[

ka 1 jka 2 jka 3 j···kanj

]T

Ifk= 0, this vector is said to be zero or null and is denoted by the boldface

symbolO.

Ifaijis a function ofx, then the derivative ofCjwith respect toxis

denoted byC

′

jand is given by the formula

C

′

j=

[

a

′

1 ja

′

2 ja

′

3 j···a

′

nj

]T

2.3 Elementary Formulas

2.3.1 Basic Properties

The arbitrary determinant

A=|aij|n=

∣

∣

C 1 C 2 C 3 ···Cn

∣

∣

,

where the suffixnhas been omitted fromAn, has the properties listed

below. Any property stated for columns can be modified to apply to rows.

a.The value of a determinant is unaltered by transposing the elements

across the principal diagonal. In symbols,

|aji|n=|aij|n.

b.The value of a determinant is unaltered by transposing the elements

across the secondary diagonal. In symbols

|an+1−j,n+1−i|n=|aij|n.

c. If any two columns ofAare interchanged and the resulting determinant

is denoted byB, thenB=−A.