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I have a question here that has no answer from the lecture notes, so I'm turning here to ask for help. The question goes like this:

Let $t_1, t_2, t_3, t_4$ represent 4 different towns in a large country. Consider that there are the following:

  • 1-way service from $t_1$ to $t_2$ and $t_3$
  • 1-way service from $t_2$ to $t_3$ and $t_4$ and
  • 2-way services between the towns $t_1$ and $t_4$

    • Write out a matrix $S$ such that $[S]_{ij} = 1$ if there is a train ride from $t_i$ to $t_j$ and $0$ otherwise.

    How will a person approach this question? I would consider a $4 \times 4$ matrix (please ignore 1st row & 1st column...it's not a $5 \times 5$ matrix).

    $$ \begin{pmatrix} * & t_1 & t_2 & t_3 & t_4 \\ t_1 & * & * & * & * \\ t_2 & * & * & * & * \\ t_3 & * & * & * & * \\ t_4 & * & * & * & * \end{pmatrix} $$

    And fill them according, resulting in

    $$S = \begin{pmatrix} \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} $$

    Is this the final answer, or should the answer be transposed?

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    • $\begingroup$ That's some strange timetable. Does the consist the has worked the 1-2-3 service return to 2 as empty stock so it can become the 2-3-4 service? And then return nonstop to 1 as a 4-1 service? How is that a meaningful service pattern? How do people who travel from 2 to 3 get back home? $\endgroup$
      – hmakholm left over Monica
      Feb 13, 2013 at 16:01
    • $\begingroup$ I had to modify the question a little to prevent Google from picking the original question up. The original question involved countries and flights. $\endgroup$
      – bryan.blackbee
      Feb 13, 2013 at 16:05
    • $\begingroup$ I would write START on the left and END at the top (or vice versa if transposed), but don't have strong feelings on which orientation is better. $\endgroup$
      – Ross Millikan
      Feb 13, 2013 at 16:14
    • $\begingroup$ Hello Ross Millikan, I am being constrained by the $ij$ part of the question, where, as you can see from the answer, is $i^{th}$ row and $j^{th}$ column. $\endgroup$
      – bryan.blackbee
      Feb 14, 2013 at 2:27

    1 Answer 1

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    The $\{ij\}^{th}$ element of the matrix will in the $i^{th}$ row and $j^{th}$ column. So your answer $$S = \begin{pmatrix} \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} $$

    is correct.

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