Let .
Find formulas for the entries of , where is a positive integer.
.
I'm not sure how to go about this. Could someone give push me in the right direction?
Probably the simplest thing to do would be to calculate what multiplying M by an arbitrary 2 x 2 matrix would give:
Now you can set up a recursion: for a = 5, b = 2, c = -1, and d = 2. This generates the next set of a, b, c, and d.
So , , etc.
Now solve the system of recursions. (That step is beyond me, I'm afraid!)
-Dan
That is not how to do it topsquark. You need to diagnolize the matrix.
Meaning,
.
Where is some invertible matrix and is a diagnol matrix. Once you do that the above equation is easy to handle.
Bring everything to power of ,
Thus,
And,
Where can be found from its eigenvectors and from its eigenvalues.
(Note, computing a power of a diagnol is easy. Just raise each diagnol entry to that exponent).