Let’s address the tasks presented in the question. We will carefully work through the solutions step by step.

5. Show that $A_1$ and $A_2$ are diagonalizable over $\mathbb{R}$:

To determine whether the matrices are diagonalizable:

1. Find the eigenvalues of each matrix by solving the characteristic equation $\det(A - \lambda I) = 0$.
2. For each eigenvalue, find the eigenspaces (i.e., the null space of $A - \lambda I$).
3. Confirm that the geometric multiplicity equals the algebraic multiplicity for each eigenvalue. If so, the matrix is diagonalizable.

6. Find invertible matrices $P_i$ ($i = 1, 2$) and diagonal matrices $D_i$ such that $A_i = P_i D_i P_i^{-1}$:

1. Using the eigenvectors found in Step 5, construct $P_i$ for each $A_1$ and $A_2$. The columns of $P_i$ are the eigenvectors.
2. Form $D_i$ as the diagonal matrix with eigenvalues of $A_i$ along the diagonal.

7. Find $P_1^{-1}$:

1. Calculate the inverse of $P_1$ from Step 6 using either the formula $P^{-1} = \text{adj}(P)/\det(P)$ or row reduction.

8. Find $A_1^n$ for all $n \in \mathbb{N}$:

1. Use the property of diagonalizable matrices: $A_1^n = P_1 D_1^n P_1^{-1}$, where $D_1^n$ is the diagonal matrix with the eigenvalues of $A_1$ raised to the $n$-th power.

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Related Questions:

1. What are the eigenvalues and eigenvectors of a $3 \times 3$ matrix like $A_1$?
2. What are the conditions for a matrix to be diagonalizable?
3. How do we compute the inverse of a matrix in general?
4. How does the diagonalization formula simplify matrix exponentiation?
5. Can non-diagonalizable matrices still have eigenvalues?

Tip:

Diagonalization is a powerful tool for simplifying computations, especially for matrix exponentiation or solving systems of differential equations.