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 $x_{0}$ .

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