Calculating the determinant of this matrix Given this (very) tricky determinant, how can we calculate it easily?
$$\begin{pmatrix} \alpha + \beta & \alpha \beta & 0 & ... & ... & 0 \\ 1 & \alpha + \beta &  \alpha \beta & 0 & ... & 0 \\ 0 & 1 & \alpha + \beta & \alpha \beta & ... & ... \\ ... & ... & ... & ... & ... & 0 \\ ... & ... & .... & ... & ... & \alpha \beta \\ 0 & 0 & 0 & ... & 1 & \alpha + \beta   \\ \end{pmatrix} \in M_{n\times n}$$
EDIT:

I have to prove it is equal to $\frac{{\alpha}^{n+1} - {\beta}^{n+1}}{\alpha - \beta}$

Any help is appreciated, I just could not find a trick to ease it up!
 A: Let $D_n$ represent the determinant of this $n\times n$ matrix.  Expanding on the first column, we see that $D_n=(\alpha+\beta)D_{n-1}-\alpha\beta D_{n-2}$, where the second is found by expanding on the first row of the resulting minor.  The recurrence begins with $D_1=\alpha+\beta, D_2=(\alpha+\beta)^2-\alpha\beta=\alpha^2+\alpha\beta+\beta^2$.
To prove that this recurrence equals $\frac{\alpha^{n+1}-\beta^{n+1}}{\alpha-\beta}$ one can use induction.
A: You can consider factorizing the matrix using L-U factorization.
Let A be your matrix, $L$ the matrix $$\begin{pmatrix} 
p_1 & 0 & 0 & ... & ... & 0 \\ 
1 & p_2 &  0 & 0 & ... & 0 \\ 
0 & 1 & p_3 & 0 & ... & ... \\ 
... & ... & ... & ... & ... & 0 \\ 
... & ... & .... & 1 & p_{n-1} & 0 \\ 
0 & 0 & 0 & ... & 1 & p_n   \\ \end{pmatrix}$$
and U the matrix
$$\begin{pmatrix} 
1 & q_1 & 0 & ... & ... & 0 \\ 
0 & 1 &  q_2 & 0 & ... & 0 \\ 
0 & 0 & 1 & q_3 & ... & ... \\ 
... & ... & ... & ... & ... & 0 \\ 
... & ... & .... & 0 & 1 & q_{n-1} \\ 
0 & 0 & 0 & ... & 0 & 1\\ \end{pmatrix}$$
where 
$p_{11} = a_{11}; \\ q_1= a_{12}/p_1; \\ p_2= a_{22} - a_{21}q_1; \\ q_2 = a_{23} / p_2; \\ p_3= a_{33}-a_{32}q_2; \\ \dots \\ \dots \\ q_{n-1}= a_{n-1,n}/p_{n-1}; \\ p_n = a_{nn} - a_{n,n-1}q_{n-1};$
then by Binet formula, you can note that $\det A = \det L \det U= \det L = \prod_{i=1}^n p_i$ 
(used the fact that $L,U$ are upper/lower triangular)
then you can use induction.
hope it helps
