Question:
Consider the following transition matrices \( P_1 \) and \( P_2 \) of two Markov chains:
\[
P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 1/3 & 1/2 & 1/6 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad P_2 = \begin{bmatrix} 1/6 & 1/3 & 1/2 \\ 1/4 & 0 & 3/4 \\ 0 & 1 & 0 \end{bmatrix}.
\]
Which of the following statements is correct?
Consider the following transition matrices \( P_1 \) and \( P_2 \) of two Markov chains:
\[
P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 1/3 & 1/2 & 1/6 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad P_2 = \begin{bmatrix} 1/6 & 1/3 & 1/2 \\ 1/4 & 0 & 3/4 \\ 0 & 1 & 0 \end{bmatrix}.
\]
Which of the following statements is correct?
Show Hint
When dealing with Markov chains, the powers of transition matrices give the probabilities of transitioning between states over multiple steps. Matrix multiplication is key to computing these powers.
Updated On: Dec 15, 2025
- \( P_1^2 = P_2 \)
- \( P_1^3 = P_2 \)
- \( P_1^4 = P_2 \)
- \( P_1^5 = P_2 \)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the transition matrices.
The transition matrix \( P_1 \) represents the transition probabilities for the first Markov chain, while \( P_2 \) represents those for the second. We need to compute powers of \( P_1 \) and compare them to \( P_2 \).
Step 2: Matrix multiplication.
To check the relationship between \( P_1 \) and \( P_2 \), we compute successive powers of \( P_1 \) and compare the results to \( P_2 \). By matrix multiplication, we find that: \[ P_1^2 = P_1 \times P_1, \quad P_1^3 = P_1 \times P_1^2, \quad P_1^4 = P_1 \times P_1^3. \] After performing the multiplications, we discover that: \[ P_1^4 = P_2. \] Thus, the correct answer is (C).
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