Let's analyze each question carefully.

Question 1

Given the dynamical system: $X_{k+1} = A X_k,$ where the matrix $A$ is: $A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}.$

To classify the origin as an attractor, repeller, saddle point, or none, we need to examine the eigenvalues of the matrix $A$.

Step 1: Find the Eigenvalues of $A$

The eigenvalues of a matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ are found by solving the characteristic polynomial: $\det(A - \lambda I) = 0.$

For our matrix: $A - \lambda I = \begin{bmatrix} 2 - \lambda & 1 \\ 0 & 3 - \lambda \end{bmatrix}.$

The determinant is: $\det(A - \lambda I) = (2 - \lambda)(3 - \lambda).$ Setting this equal to zero, we get: $(2 - \lambda)(3 - \lambda) = 0.$

So, the eigenvalues are: $\lambda_1 = 2 \quad \text{and} \quad \lambda_2 = 3.$

Step 2: Analyze the Eigenvalues

Both eigenvalues $\lambda_1 = 2$ and $\lambda_2 = 3$ are greater than 1. This means that any point $X_k$ will move away from the origin as $k \to \infty$.

Since both eigenvalues are greater than 1, the origin is classified as a repeller.

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Question 2

Compute $e^A$ for the matrix: $A = \begin{bmatrix} 1 & 2 \\ 3 & 2 \end{bmatrix}.$

To compute $e^A$, we use the matrix exponential defined by the series: $e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots.$

Step 1: Diagonalize $A$

To simplify the calculation, we can check if $A$ is diagonalizable by finding its eigenvalues and eigenvectors.

Finding Eigenvalues

1. Set up the characteristic polynomial for $A$: $\det(A - \lambda I) = 0.$

2. Substitute into the matrix