$$M = \begin{pmatrix}2&2\\ -4&8\end{pmatrix}$$
Find formulas for the entries of $M^n$, where $n$ is a positive integer.
So I think this problem involves multiplying a matrix by itself continuously. But I do not think I have observed a link between the older entries and the newer entries each time I have multiplied this matrix by itself. It just seems to be random changes in the numbers (unless I am doing something wrong). Am I supposed to create a sequence out of each of the entries and then derive a general formula?
Any assistance would be highly appreciated!
Hint: By Cayley–Hamilton, $A^2=10A-24I$. Then $A^n = a_n A + b_n I$ with $a_{n+2}=10a_{n+1}-24a_n$ and $b_{n+2}=10b_{n+1}-24b_n$. Then, both $a_n$ and $b_n$ are given by linear combinations of $4^n$ and $6^n$. The coefficients are found by considering $a_0, a_1, b_0, b_1$.