Question

i have the NxN matrix
A=[ a 1 0 . . . . . 0
1 a 1 . . . .... 0
. . . . . . . ..
0 0 0 . . 1 a 1
0 0 0 . . 0 1 a ] and i want the |A|.
I have |An|=a|An-1| - |An-2|.
can i find some more specific ?

Answer 1

This is a second-order linear recurrence (a linear difference equation)
First, you have to calculate several beginning terms (A1, A2, A3, A4, A5, etc... should be enough ...:) )
Then, you have to suppose for An the form:

An=c*x^n+d*y^n
[sum of two geometric progressions with coefficients]
When you replace you get that x and y are solutions of the characteristic equation
t^2-at+1=0;
Depending on a, you have a discussion with three cases:
solutions real, distinct (you find x and y, then c and d by using x and y and the terms A1 and A2 - the results are a bit ugly)
solutions real, equal (you use the modified general form An=c*x^n+d*n*x^n)
and you should get:
for a=2: An=n+1
for a=-2: An=(n+1)(-1)^n
solutions complex: since x and y are complex roots of a real second-order equation, they complex conjugates (and they have the complex modulus 1 and the same argument arg - the (polar) angle from their trigonometric form).
So you should again change the form you are looking for into:
An=c*sin(arg)+d*cos(arg)

You may find the general theoretic details for the above in a book about (linear) difference equations (or recurrent relations).
If you need some info, please let me know... :)