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Exercise 100 - Chapter 2

  1. Write down the matrix \( M_X \) , for \(X\) as in Eq. (2.53).

  2. Calculate \( M_X^2, M_X^3, M_X^4 \). Check that \( M_X^4 \) is what you got for the distance matrix in Exercise 2.55.

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  • 1.
    1. \[\begin{array}{c|cccc} M_X&A&B&C&D\\ \hline A&0&2&\infty&\infty\\ B&\infty&0&3&\infty\\ C&3&\infty&0&6\\ D&\infty&5&\infty&0\end{array}\]

    2. \[\begin{array}{c|cccc} M_X^2&A&B&C&D\\ \hline A&0&2&5&\infty\\ B&6&0&3&9\\ C&3&5&0&6\\ D&\infty&5&8&0\end{array}\] \[\begin{array}{c|cccc} M_X^3&A&B&C&D\\ \hline A&0&2&5&11\\ B&6&0&3&9\\ C&3&5&0&6\\ D&11&5&8&0\end{array}\] \[\begin{array}{c|cccc} M_X^4&A&B&C&D\\ \hline A&0&2&5&11\\ B&6&0&3&9\\ C&3&5&0&6\\ D&11&5&8&0\end{array}\]

    Which agrees with the matrix constructed in exercise 2.55 (transposed, since we chose opposite conventions for which side represents the start and which represents the end of the journey).

    Comment Source:1. \\[\begin{array}{c|cccc} M_X&A&B&C&D\\\\ \hline A&0&2&\infty&\infty\\\\ B&\infty&0&3&\infty\\\\ C&3&\infty&0&6\\\\ D&\infty&5&\infty&0\end{array}\\] 2. \\[\begin{array}{c|cccc} M_X^2&A&B&C&D\\\\ \hline A&0&2&5&\infty\\\\ B&6&0&3&9\\\\ C&3&5&0&6\\\\ D&\infty&5&8&0\end{array}\\] \\[\begin{array}{c|cccc} M_X^3&A&B&C&D\\\\ \hline A&0&2&5&11\\\\ B&6&0&3&9\\\\ C&3&5&0&6\\\\ D&11&5&8&0\end{array}\\] \\[\begin{array}{c|cccc} M_X^4&A&B&C&D\\\\ \hline A&0&2&5&11\\\\ B&6&0&3&9\\\\ C&3&5&0&6\\\\ D&11&5&8&0\end{array}\\] Which agrees with the matrix constructed in exercise 2.55 (transposed, since we chose opposite conventions for which side represents the start and which represents the end of the journey).
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