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The image contains a linear dynamical system problem involving a transition matrix \( T \). Here's a breakdown of the problem:

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### Question Summary:
The transition matrix \( T \) is given as:

\[
T = \begin{bmatrix} 
1.2 & 0.6 \\ 
0.3 & 0.7 
\end{bmatrix}
\]

The problem asks:

(a) To find the characteristic polynomial of \( T \) and its eigenvalues.

(b) To find the corresponding eigenvectors.

(c) To determine the long-term behavior of the system.

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### Solution:

#### Part (a) - Characteristic Polynomial and Eigenvalues:
The characteristic polynomial \( p(\lambda) \) of a matrix \( T \) is defined as:

\[
p(\lambda) = \text{det}(T - \lambda I)
\]

Where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue. We will first compute the characteristic polynomial and then solve for the eigenvalues.

Let me compute it.The eigenvalues of the matrix \( T \) are:

\[
\lambda_1 \approx 1.4424, \quad \lambda_2 \approx 0.4576
\]

These values can be obtained from the characteristic polynomial, which can be expanded further if needed.

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#### Part (b) - Eigenvectors:
Next, we will compute the eigenvectors corresponding to these eigenvalues. Let me proceed with that calculation.The eigenvectors corresponding to the eigenvalues of the matrix \( T \) are:

- Eigenvector corresponding to \( \lambda_1 \approx 1.4424 \) is:

\[
v_1 = \begin{bmatrix} 0.9272 \\ 0.3746 \end{bmatrix}
\]

- Eigenvector corresponding to \( \lambda_2 \approx 0.4576 \) is:

\[
v_2 = \begin{bmatrix} -0.6285 \\ 0.7778 \end{bmatrix}
\]

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#### Part (c) - Long-Term Behavior of the System:
The long-term behavior of a dynamical system described by a matrix depends on the eigenvalues. Specifically:

- If an eigenvalue is greater than 1, the corresponding eigenvector will grow over time.
- If an eigenvalue is less than 1, the corresponding eigenvector will decay over time.

Since \( \lambda_1 = 1.4424 > 1 \), the system will exhibit growth in the direction of the first eigenvector \( v_1 \). The second eigenvalue \( \lambda_2 = 0.4576 \), being less than 1, will cause the component along \( v_2 \) to decay over time. Therefore, the system will eventually align with the direction of the first eigenvector \( v_1 \) as time progresses.

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Would you like further details or explanations on any part?

Here are 5 related questions you might consider:

1. How do you determine stability in a dynamical system using eigenvalues?
2. What does it mean if both eigenvalues are less than 1?
3. Can eigenvalues be complex, and how do they affect the system's behavior?
4. How is the characteristic polynomial derived step by step for a 2x2 matrix?
5. How can we use diagonalization to simplify long-term behavior analysis?

**Tip:** When analyzing the long-term behavior of a system, always check the magnitude of eigenvalues to determine whether the system converges or diverges.

A linear dynamical system is described by the following transition matrix \( T \):\[ T = \begin{bmatrix} 1.2 & 0.6 \\ 0.3 & 0.7 \end{bmatrix}\] The problem asks: (a) To find the characteristic polynomial of \( T \) and its eigenvalues. (b) To find the corresponding eigenvectors. (c) To determine the long-term behavior of the system.

This problem involves calculating the characteristic polynomial, eigenvalues, and eigenvectors for a 2x2 transition matrix. It also analyzes the long-term behavior of the dynamical system based on the eigenvalues.

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