How can i get matrix from a recurrence relation?

Question

Recurrence relation: $f(n) = f(n-1) + f(n-2)$ (Fibonacci sequence)

Let, we know $k$ states of a given recurrence relation, and want to find the $(k+1)th$ state. Let $M$ be a $(k \times k)$ matrix, and we build a matrix $A:[k \times 1]$ matrix from the known states of the recurrence relation, now we want to get a matrix $B:[k \times 1]$ which will represent the set of next states, i.e. $M \times A = B$

Now i want to know the all standard rules to build matrix from a recurrence relation so that i can build matrix from any simple or complex recurrence relation.

Answer 1

Hint: Use the matrix

$$A=\begin{bmatrix} 1 & 1\\ 1 & 0 \end{bmatrix}.$$

And look at its powers $A^n$, you should see a pattern.