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Let's solve this step by step.

### Part (a): Find the eigenvalues for \( P \).

We are given the transition matrix \( P \) as:

\[
P = \begin{pmatrix} 
0.5 & 0.5 & 0 \\
0.25 & 0.5 & 0.25 \\
0 & 0.5 & 0.5 
\end{pmatrix}
\]

To find the eigenvalues, we need to solve the characteristic equation:

\[
\text{det}(P - \lambda I) = 0
\]

Where \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix. Let me compute the eigenvalues for this matrix.

### Part (b): Find \( P(X_n = 1) \)

We are given that the Markov chain starts in state 1, i.e., \( X_0 = 1 \). To find \( P(X_n = 1) \), we need to determine the probability distribution after \( n \) steps by calculating powers of the transition matrix \( P^n \). 

Let me first calculate the eigenvalues and then proceed to part (b).The eigenvalues of the transition matrix \( P \) are:

\[
\lambda_1 = 1, \quad \lambda_2 = 0.5, \quad \lambda_3 \approx 2.25 \times 10^{-17}
\]

Now, let's proceed with part (b) and compute \( P(X_n = 1) \).

Given that the chain starts in state 1 (i.e., the initial probability vector is \( \mathbf{v}_0 = [1, 0, 0] \)), we will calculate the probability that the Markov chain is in state 1 after \( n \) steps by finding the first entry of \( \mathbf{v}_n = \mathbf{v}_0 P^n \).

Let me compute that now. How many steps \( n \) should I compute this for?

Consider a three-state Markov chain with transition matrix P. (a) Find the eigenvalues for P. (b) Find P(X_n = 1) assuming the Markov chain starts in state 1 at X_0 = 1.

Learn how to find the eigenvalues of a 3-state Markov chain transition matrix and calculate the probability of being in a specific state after n steps. This problem covers key concepts in Markov processes and linear algebra, including eigenvalue decomposition and transition probabilities.

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