Recurrence relation for the determinant of a tridiagonal matrix

Question

Let

$$f_n := \begin{vmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-1} \\ & & & c_{n-1} & a_n \end{vmatrix}$$

Apparently, the determinant of the tridiagional matrix above is given by the recurrence relation

$$f_n = a_n f_{n-1} - c_{n-1} b_{n-1}f_{n-2}$$

with initial values $f_0 = 1$ and $f_{-1} = 0$ (according to Wikipedia). Can anyone please explain to me how they came to this recurrence relation? I don’t really understand how to derive it.

Answer 1

The recurrence is obtained by developing the determinant along the last column (or, equivalently, along the last row).

Answer 2

For $n \ge 2$ using Laplace expansion on the last row gives \begin{align} f_n &= \begin{vmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-3} \\ & & & c_{n-3} & a_{n-2} & b_{n-2} \\ & & & & c_{n-2} & a_{n-1} & b_{n-1} \\ & & & & & c_{n-1} & a_n \end{vmatrix} \\ &= (-1)^{2n-1} c_{n-1} \begin{vmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-3} \\ & & & c_{n-3} & a_{n-2} \\ & & & & c_{n-2} & b_{n-1} \end{vmatrix} + (-1)^{2n} a_n \begin{vmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-2} \\ & & & c_{n-2} & a_{n-1} \end{vmatrix} \\ &= - c_{n-1} (-1)^{2(n-1)} b_{n-1} \begin{vmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-3} \\ & & & c_{n-3} & a_{n-2} \end{vmatrix} + a_n f_{n-1} \\ &= a_n f_{n-1} - c_{n-1} b_{n-1} f_{n-2} \end{align} as recurrence relation.

For the initial conditions:

From comparing the above formula with matrices for $n=1$ and $n=2$ we get: $$ f_1 = a_1 f_0 - c_0 b_0 f_{-1} \overset{!}{=} a_1 \\ f_2 = a_2 f_1 - c_1 b_1 f_0 \overset{!}{=} a_1 a_2 - c_1 b_1 $$ The latter implies $f_0 = 1$ and the former $f_{-1} = 0$.