Science:Math Exam Resources/Courses/MATH307/April 2009/Question 08 (c)

MATH307 April 2009

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Question 08 (c)

A player begins a game of chance by placing a marker in box 2, marked start. A die is rolled, and the marker is moved one square to the left if a 1 or 2 is rolled and one square to the right if a 3, 4, 5, or 6 is rolled. This process continues until the marker lands in square 1, in which case the player wins the game, or in square 4, in which case the player loses the game. (c) Suppose you are told that the eigenvalues of the stochastic matrix are $\lambda _{1}=1$, $\lambda _{2}=1$, $\lambda _{3}={\frac {\sqrt {2}}{3}}$, $\lambda _{4}=-{\frac {\sqrt {2}}{3}}$, and the corresponding eigenvalues are $v_{1}={\begin{bmatrix}1\\0\\0\\0\end{bmatrix}}$, $v_{2}={\begin{bmatrix}0\\0\\0\\1\end{bmatrix}}$, $v_{3}$, $v_{4}$, and that the eigenvectors form a basis of $\mathbb {R} ^{4}$. Write down a matrix equation that you would solve in order to find the unique set of coefficients ${c_{1},c_{2},c_{3},c_{4}}$ to express the initial state of the game, $x_{0}={\begin{bmatrix}0&1&0&0\end{bmatrix}}^{T}$, in terms of the above basis of eigenvectors. Just write the equation symbolically using $v_{i}$, $c_{i}$, and $x_{0}$.

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. This becomes an $A{\vec {x}}={\vec {b}}$ problem. We want to solve for the $c_{i}$, so we start by using the vectors $v_{i}$ to build A, by writing $A={\begin{bmatrix}v_{1}&v_{2}&v_{3}&v_{4}\end{bmatrix}}$, and ${\vec {b}}={\vec {x_{0}}}$. So we can write ${\begin{bmatrix}v_{1}&v_{2}&v_{3}&v_{4}\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\\c_{4}\end{bmatrix}}={\vec {x_{0}}}$. Solving this for the $c_{i}$ would give us the set of coefficients to express the initial state of the game.