202 5. Further Determinant Theory
=
a 1 a 2 a 3 +a 1 b 3 +a 3 b 2a 1 a 2 a 3 +a 1 b 3 +a 3 b 2 +a 2 a 3 b 1 +b 1 b 3.
Each of these fractions can be expressed in the formH 11 /H, whereHis a
tridiagonal determinant:
f 1 =|a 1 |
∣
∣
∣
∣1 b 1− 1 a 1∣
∣
∣
∣
,
f 2 =∣
∣
∣
∣
a 1 b 2− 1 a 2∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
1 b 1− 1 a 1 b 2− 1 a 2∣ ∣ ∣ ∣ ∣ ∣
,
f 3 =∣ ∣ ∣ ∣ ∣ ∣
a 1 b 2− 1 a 2 b 3− 1 a 3∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
1 b 1− 1 a 1 b 2− 1 a 2 b 3− 1 a 3∣ ∣ ∣ ∣ ∣ ∣ ∣
.
Theorem 5.9.
fn=H
(n+1)
11Hn+1,
where
Hn+1=∣
∣
∣
∣
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
1 b 1− 1 a 1 b 2− 1 a 2 b 3.
.
..
.
.
.
.
.
− 1 an− 2 bn− 1− 1 an− 1 bn− 1 an∣
∣
∣
∣
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
n+1. (5.5.2)
Proof. Use the method of induction. Assume that
fn− 1 =H
(n)
11Hn,
which is known to be true for small values ofn. Hence, addingbn/anto
an− 1 ,
fn=K
(n)
11Kn