Henry Ernest Dudeney/Modern Puzzles/22 - Mrs. Wilson's Family/Working

Working for Modern Puzzles by Henry Ernest Dudeney: $22$ -- Mrs. Wilson's Family

The simultaneous equations in matrix form:

$\begin {pmatrix}

1 & 0 & -1 & 1 & -1 & 0 \\ 1 & 1 & 1 & 1 & 1 & -2 \\ 1 & 1 & 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 & -1 & 0 \\ 1 & 1 & 1 & 0 & 1 & -1 \\ 2 & 2 & 2 & 0 & 0 & -1 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \\ M \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 30 \\ 75 \\ \end {pmatrix}$

when converted to reduced echelon form, gives:

$\begin {pmatrix}

1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \\ M \end {pmatrix} = \begin {pmatrix} 21 \\ 18 \\ 18 \\ 9 \\ 12 \\ 39 \\ \end {pmatrix}$

Proof

\(\ds \)

\(\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & -2 & 0 \\ 1 & 1 & 0 & 0 & 0 & -1 & 0 \\ 1 & 0 & 0 & -1 & -1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & -1 & 30 \\ 2 & 2 & 2 & 0 & 0 & -1 & 75 \\ \end {array} }\)

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 1 & 1 & -1 & 1 & -1 & 0 \\ 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 1 & 2 & -1 & 2 & -1 & 30 \\ 0 & 2 & 4 & -2 & 2 & -1 & 75 \\ \end {array} }\)

$r_2 \to r_2 - r_1$, $r_3 \to r_3 - r_1$, $r_4 \to r_4 - r_1$, $r_5 \to r_5 - r_1$, $r_6 \to r_6 - 2 r_1$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & -1 & -1 & -1 & 1 & 0 \\ 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 1 & 30 \\ 0 & 0 & 0 & -2 & -2 & 3 & 75 \\ \end {array} }\)

$r_3 \to r_3 - r_2$, $r_5 \to r_5 - r_2$, $r_6 \to r_6 - 2 r_2$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 1 & 30 \\ 0 & 0 & 0 & -2 & -2 & 3 & 75 \\ \end {array} }\)

$r_3 \to -r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & -3 & -1 & 1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 1 & 30 \\ 0 & 0 & 0 & -2 & -2 & 3 & 75 \\ \end {array} }\)

$r_4 \to r_4 - r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 1 & 30 \\ 0 & 0 & 0 & -3 & -1 & 1 & 0 \\ 0 & 0 & 0 & -2 & -2 & 3 & 75 \\ \end {array} }\)

$r_4 \leftrightarrow r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 & -30 \\ 0 & 0 & 0 & -3 & -1 & 1 & 0 \\ 0 & 0 & 0 & -2 & -2 & 3 & 75 \\ \end {array} }\)

$r_4 \to -r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 & -30 \\ 0 & 0 & 0 & 0 & -1 & -2 & -90 \\ 0 & 0 & 0 & 0 & -2 & 1 & 15 \\ \end {array} }\)

$r_5 \to r_5 + 3 r_4$, $r_6 \to r_6 + 2 r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 & -30 \\ 0 & 0 & 0 & 0 & 1 & 2 & 90 \\ 0 & 0 & 0 & 0 & -2 & 1 & 15 \\ \end {array} }\)

$r_5 \to -r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 & -30 \\ 0 & 0 & 0 & 0 & 1 & 2 & 90 \\ 0 & 0 & 0 & 0 & 0 & 5 & 195 \\ \end {array} }\)

$r_6 \to r_6 + 2 r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & -2 & 0 \\ 0 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 & -30 \\ 0 & 0 & 0 & 0 & 1 & 2 & 90 \\ 0 & 0 & 0 & 0 & 0 & 1 & 39 \\ \end {array} }\)

$r_6 \to r_6 / 5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 2 & 0 & 78 \\ 0 & 0 & 1 & 1 & 1 & 0 & 39 \\ 0 & 0 & 0 & 1 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 0 & 1 & 39 \\ \end {array} }\)

$r_5 \to r_5 - 2 r_6$, $r_4 \to r_4 + r_6$, $r_3 \to r_3 + r_6$, $r_2 \to r_2 + 2 r_6$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 1 & 0 & 0 & 12 \\ 0 & 1 & 2 & 0 & 0 & 0 & 54 \\ 0 & 0 & 1 & 1 & 0 & 0 & 27 \\ 0 & 0 & 0 & 1 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 0 & 1 & 39 \\ \end {array} }\)

$r_3 \to r_3 - r_5$, $r_2 \to r_2 - 2 r_5$, $r_1 \to r_1 + r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & -1 & 0 & 0 & 0 & 3 \\ 0 & 1 & 2 & 0 & 0 & 0 & 54 \\ 0 & 0 & 1 & 0 & 0 & 0 & 18 \\ 0 & 0 & 0 & 1 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 0 & 1 & 39 \\ \end {array} }\)

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

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & 0 & 21 \\ 0 & 1 & 0 & 0 & 0 & 0 & 18 \\ 0 & 0 & 1 & 0 & 0 & 0 & 18 \\ 0 & 0 & 0 & 1 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 0 & 1 & 39 \\ \end {array} }\)

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

$\blacksquare$