Page 29 - Determinants and Their Applications in Mathematical Physics

14 2. A Summary of Basic Determinant Theory

1
n
= b i A ij . (2.3.14)
A
i=1
The solution of the triangular set of equations
i

a ij x j = b i , i =1, 2, 3,...
j=1
(the upper limit in the sum is i, not n as in the previous set) is given by
the formula

b 1 a 11

b 2 a 21 a 22
(−1) i+1 b 3 a 31 a 32 a 33
x i = .

a 11 a 22 ··· a ii ··· ··· ··· ··· ··· ···

b i−1 a i−1,1 a i−1,2 a i−1,3 ··· a i−1,i−1

a i1 a i2 a i3 ··· a i,i−1
b i
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
n
(n)
f = a ij x j + a in , 1 ≤ i ≤ n,
i
j=1
and
(n)
f =0, 1 ≤ i ≤ n, i = r,
i
prove that
f (n) = A n x r , 1 ≤ r< n,
(n)
r A rn
A n (x n +1)
(n)
f = ,
n
A n−1
where
A n = |a ij | n ,
provided
A (n) =0, 1 ≤ i ≤ n.
rn