Determinants and Their Applications in Mathematical Physics

14 2. A Summary of Basic Determinant Theory

=

1

A

n

∑

i=1

biAij. (2.3.14)

The solution of the triangular set of equations

i

∑

j=1

aijxj=bi,i=1, 2 , 3 ,...

(the upper limit in the sum isi, notnas in the previous set) is given by

the formula

xi=

(−1)

i+1

a 11 a 22 ···aii

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

b 1 a 11

b 2 a 21 a 22

b 3 a 31 a 32 a 33

··· ··· ··· ··· ··· ···

bi− 1 ai− 1 , 1 ai− 1 , 2 ai− 1 , 3 ··· ai− 1 ,i− 1

bi ai 1 ai 2 ai 3 ··· ai,i− 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

i

.

(2.3.15)

The determinant is a Hessenbergian (Section 4.6).

Cramer’s formula is of great theoretical interest and importance in solv-

ing sets of equations with algebraic coefficients but is unsuitable for reasons

of economy for the solution of large sets of equations with numerical coeffi-

cients. It demands far more computation than the unavoidable minimum.

Some matrix methods are far more efficient. Analytical applications of

Cramer’s formula appear in Section 5.1.2 on the generalized geometric se-

ries, Section 5.5.1 on a continued fraction, and Section 5.7.2 on the Hirota

operator.

Exercise.If

f

(n)

i

=

n

∑

j=1

aijxj+ain, 1 ≤i≤n,

and

f

(n)

i

=0, 1 ≤i≤n, i=r,

prove that

f

(n)

r

=

Anxr

A

(n)

rn

, 1 ≤r<n,

f

(n)

n =

An(xn+1)

An− 1

,

where

An=|aij|n,

provided

A

(n)

rn =0,^1 ≤i≤n.