Problem Solving for finding an expression for $A^n$

Question

Problem: For $A=\begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \in M_{2 \times 2}(R)$, find an expression for $A^n$, where n is an arbitrary positive integer.

Question 1: I was told I need to find eigenvalues to test the whether the matrix is diagonalizable or not? If that is diagonolizable, then we can apply the formula $A=PDP^{-1}$, where D is the diagonal matrix for A to find the expression of $A^n$? If not diagonolizable, we can't use that formula right?

Question 2: We say a matrix A is diagonalizable to say that we can always find a diagonal matrix D such that $A=PDP^{-1}$, where P is an invertible matrix?

Answer 1

Yes, upon finding the eigenvalue and eigenvector pair.

We have $$A^n=PD^nP^{-1}$$

Here $D^n$ can be computed easily as $D$ is a diagonal matrix and we can just raised the component separately.

Remark: The trace of the matrix is $4$ and the determinant is $-5$. Hence the eigenvalues are $5$ and $-1$. Since the eigenvalues are distinct, $A$ is diagonalizable.

Edit:

Diagonalizable means there exists an invertible $P$ such that $A=PDP^{-1}$ where $D$ is a diagonal matrix.