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

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.

## Comments

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

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

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).

`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).`