Henry Ernest Dudeney/Puzzles and Curious Problems/75 - A Question of Transport/Working

== Working for by : $75$ -- A Question of Transport ==

The system of simultaneous equations in matrix form:


 * $\begin {pmatrix}

1 & 0 &  0 &  0 & -20 &   0 &   0 &   0 &  0 \\ 0 &  1 &  0 &  0 &   0 &  -4 &   0 &   0 &  0 \\ 1 & -1 &  0 &  0 &  20 & -20 &   0 &   0 &  0 \\ -1 &  0 &  1 &  0 &   4 &   0 &  -4 &   0 &  0 \\ 0 & -1 &  1 &  0 &   0 &  20 & -20 &   0 &  0 \\ 0 & -1 &  0 &  1 &   0 &   4 &   0 &  -4 &  0 \\ 0 &  0 &  1 & -1 &   0 &   0 &  20 & -20 &  0 \\ 0 &  0 &  1 &  0 &   0 &   0 &  -4 &   0 &  4 \\ 0 &  0 &  0 &  1 &   0 &   0 &   0 & -20 & 20 \\ \end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ t_1 \\ t_2 \\ t_3 \\ t_4 \\ t_5 \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 20 \\ 20 \end {pmatrix}$

when converted to echelon form, gives:
 * $\begin {pmatrix}

1 & 0 &  0 &  0 & -20 &   0 &    0 &    0 &    0 \\ 0 &  1 &  0 &  0 &   0 &  -4 &    0 &    0 &    0 \\ 0 &  0 &  1 &  0 & -16 &   0 &   -4 &    0 &    0 \\ 0 &  0 &  0 &  1 &   0 &   0 &    0 &   -4 &    0 \\ 0 &  0 &  0 &  0 &   1 &   1 &   -1 &    0 &    0 \\ 0 &  0 &  0 &  0 &   0 &   1 & -5/2 &  3/2 &    0 \\ 0 &  0 &  0 &  0 &   0 &   0 &    1 &   -1 & -1/6 \\ 0 &  0 &  0 &  0 &   0 &   0 &    0 &    1 &  5/6 \\ 0 &  0 &  0 &  0 &   0 &   0 &    0 &    0 &    1 \\ \end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ t_1 \\ t_2 \\ t_3 \\ t_4 \\ t_5 \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ -5/6 \\ 25/6 \\ 13/5 \end {pmatrix}$