Page 23 - Determinants and Their Applications in Mathematical Physics

8 2. A Summary of Basic Determinant Theory

R 1

R 2

A n = R 3 = C 1 C 2 C 3 ··· C n . (2.2.2)

.
.
.
R n
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 symbols R i and C j
appear with equal frequency.
If every element in C j is multiplied by the scalar k, the resulting vector
is denoted by kC j :

.
T
kC j = ka 1j ka 2j ka 3j ··· ka nj
If k = 0, this vector is said to be zero or null and is denoted by the boldface
symbol O.
If a ij is a function of x, then the derivative of C j with respect to x is
denoted by C and is given by the formula

j
T

C = a a a ··· a .

j 1j 2j 3j nj
2.3 Elementary Formulas

2.3.1 Basic Properties

The arbitrary determinant

A = |a ij | n = C 1 C 2 C 3 ··· C n ,

where the suffix n has been omitted from A n , 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,

|a ji | n = |a ij | n .
b. The value of a determinant is unaltered by transposing the elements
across the secondary diagonal. In symbols
|a n+1−j,n+1−i | n = |a ij | n .

c. If any two columns of A are interchanged and the resulting determinant
is denoted by B, then B = −A.