Science:Math Exam Resources/Courses/MATH307/April 2010/Question 06 (a)

MATH307 April 2010

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Question 06 (a)
Consider the matrix $P={\begin{bmatrix}{\frac {1}{2}}&0&0&0\\{\frac {1}{6}}&\alpha &0&*\\{\frac {1}{6}}&0&\beta &0\\{\frac {1}{6}}&*&*&\gamma \end{bmatrix}}$ Fill in the star entries so that P is a stochastic matrix.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. For P to be a stochastic matrix, all entries must be non-negative, and all entries in each column must sum to 1. The first column already satisfies this requirement. For the second column, it must be that $0+\alpha +0+*=1$. So, it follows that $*=1-\alpha$. For the third column, it must be that $0+0+\beta +*=1$. So, it follows that $*=1-\beta$. For the fourth column, it must be that $0+*+0+\gamma =1$. So, it follows that $*=1-\gamma$. Filling in the star entries, the stochastic matrix is: $P={\begin{bmatrix}{\frac {1}{2}}&0&0&0\\{\frac {1}{6}}&\alpha &0&1-\gamma \\{\frac {1}{6}}&0&\beta &0\\{\frac {1}{6}}&1-\alpha &1-\beta &\gamma \end{bmatrix}}$