Solution of a linear system of difference equations

Question

In the discrete case of dynamical systems, we are interested with how points in space behave under iterations by a particular map, $f$. If we restrict ourselves to the plane, and only consider the case when $f$ is linear, we can represent the mapping by multiplication with a coefficient matrix $A$. Since maps are iterative, we have, in general, that $$p_{n+1} = A^{n + 1} p_0 = Ap_n$$ where $p_i$ is any point in the plane after $i$ iterations. In general, how do we reach the general solution of such a system given by $X_{n+1} = AX_n$, for $A \in \mathbb{M}_{2\times2}$ and $X_i \in \mathbb{R}^2$ for any $i$? I've had trouble finding concrete answers online, and the YouTube videos I've found have offered little solace. For the sake of clarity, let $\displaystyle A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$.