I am trying to solve a research problem, and I am stuck at this point. I need to find the determinant of the following matrix.
Let $a_1, \dots, a_p > 0$ and $b_1, \dots, b_{p-1} > 0$. Define the tridiagonal $p \times p$ symmetric matrix as
$$\text{Main diagonals:} \quad \left \{a_i + b_{i-1} + b_i, \; i = 1, \dots, p \right\} $$ $$\text{Off diagonals:} \quad \left \{ -b_i, \; i = 1, \dots, p-1 \right\}, $$
So the matrix $\Sigma$ looks like. $$\Sigma = \left[\begin{array}{ccccc} a_1 + b_1 & -b_1 & 0 & \dots & \dots \\ -b_1 & a_2 + b_1 + b_2 & -b_2 & \dots & \dots\\ 0 & -b_2 & a_3 + b_2 + b_3 & & \dots\\\dots & \dots & \dots & \ddots & \dots \\ \dots & \dots & \dots & \dots & -b_{p-1} \\ \dots & \dots & \dots & -b_{p-1} & a_p + b_{p-1}\end{array} \right].$$
Is there a closed form expression for the determinant of $\Sigma$?