Henry Ernest Dudeney/Puzzles and Curious Problems/2 - A Legacy Puzzle/Working

Working for Puzzles and Curious Problems by Henry Ernest Dudeney: $2$ -- A Legacy Puzzle

The simultaneous equations in matrix form:

$\quad \begin {pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & -1 & 1 \\ -2 & 1 & -2 & 1 \\ -3 & -3 & 1 & 1 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ h \end {pmatrix} = \begin {pmatrix} 1320 \\ 0 \\ 0 \\ 0 \end {pmatrix}$

when converted to reduced echelon form, gives:

$\quad \begin {pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ h \end {pmatrix} = \begin {pmatrix} 55 \\ 275 \\ 385 \\ 605 \\ \end {pmatrix}$

Proof

\(\ds \)

\(\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 1 & 1320 \\ 1 & -1 & -1 & 1 & 0 \\ -2 & 1 & -2 & 1 & 0 \\ -3 & -3 & 1 & 1 & 0 \\ \end {array} }\)

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 1 & 1320 \\ 0 & -2 & -2 & 0 & -1320 \\ 0 & 3 & 0 & 3 & 2640 \\ 0 & 0 & 4 & 4 & 3960 \\ \end {array} }\)

$r_2 \to r_2 - r_1$, $r_3 \to r_3 + 2 r_1$, $r_4 \to r_4 + 3 r_1$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 1 & 1320 \\ 0 & 1 & 1 & 0 & 660 \\ 0 & 1 & 0 & 1 & 880 \\ 0 & 0 & 1 & 1 & 990 \\ \end {array} }\)

$r_2 \to -\dfrac {r_2} 2$, $r_3 \to \dfrac {r_3} 3$, $r_4 \to \dfrac {r_4} 4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 1 & 1320 \\ 0 & 1 & 1 & 0 & 660 \\ 0 & 0 & -1 & 1 & 220 \\ 0 & 0 & 1 & 1 & 990 \\ \end {array} }\)

$r_3 \to r_3 - r_2$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 1 & 1320 \\ 0 & 1 & 1 & 0 & 660 \\ 0 & 0 & 1 & -1 & -220 \\ 0 & 0 & 1 & 1 & 990 \\ \end {array} }\)

$r_3 \to -r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 1 & 1320 \\ 0 & 1 & 1 & 0 & 660 \\ 0 & 0 & 1 & -1 & -220 \\ 0 & 0 & 0 & 2 & 1210 \\ \end {array} }\)

$r_4 \to r_4 - r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 1 & 1320 \\ 0 & 1 & 1 & 0 & 660 \\ 0 & 0 & 1 & -1 & -220 \\ 0 & 0 & 0 & 1 & 605 \\ \end {array} }\)

$r_4 \to \dfrac {r_4} 2$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 1 & 1 & 0 & 715 \\ 0 & 1 & 1 & 0 & 660 \\ 0 & 0 & 1 & 0 & 385 \\ 0 & 0 & 0 & 1 & 605 \\ \end {array} }\)

$r_3 \to r_3 + r_4$, $r_1 \to r_1 - r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 0 & 0 & 0 & 55 \\ 0 & 1 & 1 & 0 & 660 \\ 0 & 0 & 1 & 0 & 385 \\ 0 & 0 & 0 & 1 & 605 \\ \end {array} }\)

$r_1 \to r_1 - r_2$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccc{{|}}c} 1 & 0 & 0 & 0 & 55 \\ 0 & 1 & 0 & 0 & 275 \\ 0 & 0 & 1 & 0 & 385 \\ 0 & 0 & 0 & 1 & 605 \\ \end {array} }\)

$r_2 \to r_2 - r_3$

$\blacksquare$