Henry Ernest Dudeney/Modern Puzzles/36 - More Bicycling/Working

Working for Modern Puzzles by Henry Ernest Dudeney: $36$ -- More Bicycling

The simultaneous equations in matrix form:

$\begin {pmatrix}

1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & -8 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & -3 & 0 \\ 1 & 1 & -4 & 0 & 0 & -4 & 0 & 0 & 4 \\ 1 & 1 & 0 & -5 & 0 & 0 & -5 & 0 & 5 \\ 1 & 1 & 0 & 0 & -12 & 0 & 0 & -12 & 12 \\ \end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ t_{a_1} \\ t_{b_1} \\ t_{c_1} \\ t_{a_2} \\ t_{b_2} \\ t_{c_2} \\ t \\ \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 20 \\ 20 \\ 20 \\ \end {pmatrix}$

when converted to reduced echelon form, gives:

$\begin {pmatrix}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ t_{a_1} \\ t_{b_1} \\ t_{c_1} \\ t_{a_2} \\ t_{b_2} \\ t_{c_2} \\ t \\ \end {pmatrix} = \begin {pmatrix} 200/27 \\ 40/27 \\ 20/27 \\ 40/27 \\ 200/81 \\ 10/27 \\ 5/27 \\ 40/81 \\ 35/9 \\ \end {pmatrix}$

Proof

\(\ds \)

\(\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 \\ 1 & 1 & -4 & 0 & 0 & -4 & 0 & 0 & 4 & 20 \\ 1 & 1 & 0 & -5 & 0 & 0 & -5 & 0 & 5 & 20 \\ 1 & 1 & 0 & 0 & -12 & 0 & 0 & -12 & 12 & 20 \\ \end {array} }\)

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10 & -5 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10 & 0 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 \\ 0 & 1 & 6 & 0 & 0 & -4 & 0 & 0 & 4 & 20 \\ 0 & 1 & 10 & -5 & 0 & 0 & -5 & 0 & 5 & 20 \\ 0 & 1 & 10 & 0 & -12 & 0 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_2 \to r_2 - r_1$, $r_3 \to r_3 - r_1$, $r_7 \to r_7 - r_1$, $r_8 \to r_8 - r_1$, $r_9 \to r_9 - r_1$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10 & -5 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10 & 0 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 \\ 0 & 1 & 6 & 0 & 0 & -4 & 0 & 0 & 4 & 20 \\ 0 & 1 & 10 & -5 & 0 & 0 & -5 & 0 & 5 & 20 \\ 0 & 1 & 10 & 0 & -12 & 0 & 0 & -12 & 12 & 20 \\ \end {array} }\)

rearranging rows for convenience

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10 & -5 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10 & 0 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 10 & -5 & 0 & 4 & -5 & 0 & 5 & 20 \\ 0 & 0 & 10 & 0 & -12 & 4 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_5 \to r_5 - r_2$, $r_6 \to r_6 - r_2$, $r_7 \to r_7 - r_2$, $r_8 \to r_8 - r_2$, $r_9 \to r_9 - r_2$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10 & -5 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ 0 & 0 & 0 & 5 & -12 & 4 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_4 \to r_4 - r_3$, $r_8 \to r_8 - r_3$, $r_9 \to r_9 - r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ 0 & 0 & 0 & 5 & -12 & 4 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_3 \to r_3 / 10$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ 0 & 0 & 0 & 5 & -12 & 4 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_7 \to r_7 - 6 r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 & -3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_9 \to r_9 - r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_4 \to r_4 / 5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 9/5 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ \end {array} }\)

$r_7 \to r_7 - 3 r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 9/5 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ \end {array} }\)

rearranging for convenience

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 20 & 100 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ \end {array} }\)

$r_8 \to r_8 \times 5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & -12 & 32 & 120 \\ 0 & 0 & 0 & 0 & 0 & 4 & -5 & 0 & 5 & 20 \\ \end {array} }\)

$r_8 \to r_8 + r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & -12 & 32 & 120 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 5 & 20 \\ \end {array} }\)

$r_7 \to r_7 - r_6$, $r_8 \to r_8 - r_6$, $r_9 \to r_9 - r_6$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 4 & -8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & -12 & 32 & 120 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 5 & 20 \\ \end {array} }\)

rearranging for convenience

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -3/2 & 4 & 15 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 5 & 20 \\ \end {array} }\)

$r_6 \to r_6 / 4$, $r_7 \to r_7 / 8$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -3/2 & 4 & 15 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & -32 & -120 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9/2 & -7 & -25 \\ \end {array} }\)

$r_8 \to r_8 - 8 r_7$, $r_9 \to r_9 - 3 r_7$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -3/2 & 4 & 15 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & -32 & -120 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & -14 & -50 \\ \end {array} }\)

$r_9 \to r_9 \times 2$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -9 & 4 & 0 & -12 & 12 & 20 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -3/2 & 4 & 15 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & -32 & -120 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 18 & 70 \\ \end {array} }\)

$r_9 \to r_9 - r_8$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -4/9 & 0 & 4/3 & -4/3 & -20/9 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -3/2 & 4 & 15 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -32/9 & -40/3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_9 \to r_9 / 18$, $r_8 \to r_8 / 9$, $r_5 \to r_5 / \paren {-9}$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -4/9 & 0 & 4/3 & 0 & 80/27 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -3/2 & 0 & -5/9 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 40/81 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_8 \to r_8 + \dfrac {32} 9 r_9$, $r_7 \to r_7 - 4 r_9$, $r_5 \to r_5 + \dfrac 4 3 r_9$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -4/9 & 0 & 0 & 0 & 560/243 \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 5/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 40/81 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_7 \to r_7 + \dfrac 3 2 r_8$, $r_5 \to r_5 - \dfrac 4 3 r_8$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -4/9 & 0 & 0 & 0 & 560/243 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 10/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 5/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 40/81 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_6 \to r_6 + 2 r_7$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 40/27 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3/5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 200/81 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 10/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 5/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 40/81 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_5 \to r_5 + \dfrac 4 9 r_6$, $r_2 \to r_2 + \dfrac 4 r_6$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 40/27 \\ 0 & 0 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 40/27 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 200/81 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 10/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 5/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 40/81 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_4 \to r_4 + \dfrac 3 5 r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 40/27 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 20/27 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 40/27 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 200/81 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 10/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 5/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 40/81 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_3 \to r_3 + \dfrac 1 2 r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 200/27 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 40/27 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 20/27 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 40/27 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 200/81 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 10/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 5/27 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 40/81 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 35/9 \\ \end {array} }\)

$r_1 \to r_1 + 10 r_3$

$\blacksquare$