Consider the dynamical system Xk+1 = AXk where 

2 1

0 3

. Classify the origin as an attractor,

repeller, saddle point or none of these.

Question 2

Compute e

A

for 

1 2

3 2

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**.

---

### 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

Consider the dynamical system Xk+1 = AXk where A = [[2, 1], [0, 3]]. Classify the origin as an attractor, repeller, saddle point, or none of these.

Question 2

Compute e^A for A = [[1, 2], [3, 2]]

Explore how to classify the origin in a dynamical system with matrix A = [[2, 1], [0, 3]] as an attractor, repeller, or saddle point by analyzing eigenvalues. Additionally, learn the method to compute the matrix exponential e^A for A = [[1, 2], [3, 2]]. Ideal for college-level studies in linear algebra and dynamical systems.

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