Конечная математика Примеры

$[\begin{array}{c} 36 & 50 & 48 & 29 \\ 5 & 6 & 7 & 8 \\ 2 & 1 & 3 & 4 \\ 4 & 2 & 3 & 1 \end{array}]$

Этап 1

Multiply each element of $R_{1}$ by $\frac{1}{36}$ to make the entry at $1, 1$ a $1$ .

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Этап 1.1

Multiply each element of $R_{1}$ by $\frac{1}{36}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{36}{36} & \frac{50}{36} & \frac{48}{36} & \frac{29}{36} \\ 5 & 6 & 7 & 8 \\ 2 & 1 & 3 & 4 \\ 4 & 2 & 3 & 1 \end{array}]$

Этап 1.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 5 & 6 & 7 & 8 \\ 2 & 1 & 3 & 4 \\ 4 & 2 & 3 & 1 \end{array}]$

Этап 2

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Этап 2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 5 - 5 \cdot 1 & 6 - 5 (\frac{25}{18}) & 7 - 5 (\frac{4}{3}) & 8 - 5 (\frac{29}{36}) \\ 2 & 1 & 3 & 4 \\ 4 & 2 & 3 & 1 \end{array}]$

Этап 2.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & - \frac{17}{18} & \frac{1}{3} & \frac{143}{36} \\ 2 & 1 & 3 & 4 \\ 4 & 2 & 3 & 1 \end{array}]$

Этап 3

Perform the row operation $R_{3} = R_{3} - 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Этап 3.1

Perform the row operation $R_{3} = R_{3} - 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & - \frac{17}{18} & \frac{1}{3} & \frac{143}{36} \\ 2 - 2 \cdot 1 & 1 - 2 (\frac{25}{18}) & 3 - 2 (\frac{4}{3}) & 4 - 2 (\frac{29}{36}) \\ 4 & 2 & 3 & 1 \end{array}]$

Этап 3.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & - \frac{17}{18} & \frac{1}{3} & \frac{143}{36} \\ 0 & - \frac{16}{9} & \frac{1}{3} & \frac{43}{18} \\ 4 & 2 & 3 & 1 \end{array}]$

Этап 4

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Этап 4.1

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

Этап 4.2

Упростим $R_{4}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & - \frac{17}{18} & \frac{1}{3} & \frac{143}{36} \\ 0 & - \frac{16}{9} & \frac{1}{3} & \frac{43}{18} \\ 0 & - \frac{32}{9} & - \frac{7}{3} & - \frac{20}{9} \end{array}]$

Этап 5

Multiply each element of $R_{2}$ by $- \frac{18}{17}$ to make the entry at $2, 2$ a $1$ .

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Этап 5.1

Multiply each element of $R_{2}$ by $- \frac{18}{17}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ - \frac{18}{17} \cdot 0 & - \frac{18}{17} (- \frac{17}{18}) & - \frac{18}{17} \cdot \frac{1}{3} & - \frac{18}{17} \cdot \frac{143}{36} \\ 0 & - \frac{16}{9} & \frac{1}{3} & \frac{43}{18} \\ 0 & - \frac{32}{9} & - \frac{7}{3} & - \frac{20}{9} \end{array}]$

Этап 5.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & - \frac{16}{9} & \frac{1}{3} & \frac{43}{18} \\ 0 & - \frac{32}{9} & - \frac{7}{3} & - \frac{20}{9} \end{array}]$

Этап 6

Perform the row operation $R_{3} = R_{3} + \frac{16}{9} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Этап 6.1

Perform the row operation $R_{3} = R_{3} + \frac{16}{9} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 + \frac{16}{9} \cdot 0 & - \frac{16}{9} + \frac{16}{9} \cdot 1 & \frac{1}{3} + \frac{16}{9} (- \frac{6}{17}) & \frac{43}{18} + \frac{16}{9} (- \frac{143}{34}) \\ 0 & - \frac{32}{9} & - \frac{7}{3} & - \frac{20}{9} \end{array}]$

Этап 6.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & - \frac{5}{17} & - \frac{173}{34} \\ 0 & - \frac{32}{9} & - \frac{7}{3} & - \frac{20}{9} \end{array}]$

Этап 7

Perform the row operation $R_{4} = R_{4} + \frac{32}{9} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Этап 7.1

Perform the row operation $R_{4} = R_{4} + \frac{32}{9} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & - \frac{5}{17} & - \frac{173}{34} \\ 0 + \frac{32}{9} \cdot 0 & - \frac{32}{9} + \frac{32}{9} \cdot 1 & - \frac{7}{3} + \frac{32}{9} (- \frac{6}{17}) & - \frac{20}{9} + \frac{32}{9} (- \frac{143}{34}) \end{array}]$

Этап 7.2

Упростим $R_{4}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & - \frac{5}{17} & - \frac{173}{34} \\ 0 & 0 & - \frac{61}{17} & - \frac{292}{17} \end{array}]$

Этап 8

Multiply each element of $R_{3}$ by $- \frac{17}{5}$ to make the entry at $3, 3$ a $1$ .

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Этап 8.1

Multiply each element of $R_{3}$ by $- \frac{17}{5}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ - \frac{17}{5} \cdot 0 & - \frac{17}{5} \cdot 0 & - \frac{17}{5} (- \frac{5}{17}) & - \frac{17}{5} (- \frac{173}{34}) \\ 0 & 0 & - \frac{61}{17} & - \frac{292}{17} \end{array}]$

Этап 8.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & 1 & \frac{173}{10} \\ 0 & 0 & - \frac{61}{17} & - \frac{292}{17} \end{array}]$

Этап 9

Perform the row operation $R_{4} = R_{4} + \frac{61}{17} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Этап 9.1

Perform the row operation $R_{4} = R_{4} + \frac{61}{17} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & 1 & \frac{173}{10} \\ 0 + \frac{61}{17} \cdot 0 & 0 + \frac{61}{17} \cdot 0 & - \frac{61}{17} + \frac{61}{17} \cdot 1 & - \frac{292}{17} + \frac{61}{17} \cdot \frac{173}{10} \end{array}]$

Этап 9.2

Упростим $R_{4}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & 1 & \frac{173}{10} \\ 0 & 0 & 0 & \frac{449}{10} \end{array}]$

Этап 10

Multiply each element of $R_{4}$ by $\frac{10}{449}$ to make the entry at $4, 4$ a $1$ .

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Этап 10.1

Multiply each element of $R_{4}$ by $\frac{10}{449}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & 1 & \frac{173}{10} \\ \frac{10}{449} \cdot 0 & \frac{10}{449} \cdot 0 & \frac{10}{449} \cdot 0 & \frac{10}{449} \cdot \frac{449}{10} \end{array}]$

Этап 10.2

Упростим $R_{4}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & 1 & \frac{173}{10} \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 11

Perform the row operation $R_{3} = R_{3} - \frac{173}{10} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Этап 11.1

Perform the row operation $R_{3} = R_{3} - \frac{173}{10} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 - \frac{173}{10} \cdot 0 & 0 - \frac{173}{10} \cdot 0 & 1 - \frac{173}{10} \cdot 0 & \frac{173}{10} - \frac{173}{10} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 11.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & - \frac{143}{34} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 12

Perform the row operation $R_{2} = R_{2} + \frac{143}{34} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Этап 12.1

Perform the row operation $R_{2} = R_{2} + \frac{143}{34} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 + \frac{143}{34} \cdot 0 & 1 + \frac{143}{34} \cdot 0 & - \frac{6}{17} + \frac{143}{34} \cdot 0 & - \frac{143}{34} + \frac{143}{34} \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 12.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & \frac{29}{36} \\ 0 & 1 & - \frac{6}{17} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 13

Perform the row operation $R_{1} = R_{1} - \frac{29}{36} R_{4}$ to make the entry at $1, 4$ a $0$ .

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Этап 13.1

Perform the row operation $R_{1} = R_{1} - \frac{29}{36} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{29}{36} \cdot 0 & \frac{25}{18} - \frac{29}{36} \cdot 0 & \frac{4}{3} - \frac{29}{36} \cdot 0 & \frac{29}{36} - \frac{29}{36} \cdot 1 \\ 0 & 1 & - \frac{6}{17} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 13.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & 0 \\ 0 & 1 & - \frac{6}{17} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 14

Perform the row operation $R_{2} = R_{2} + \frac{6}{17} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Этап 14.1

Perform the row operation $R_{2} = R_{2} + \frac{6}{17} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & 0 \\ 0 + \frac{6}{17} \cdot 0 & 1 + \frac{6}{17} \cdot 0 & - \frac{6}{17} + \frac{6}{17} \cdot 1 & 0 + \frac{6}{17} \cdot 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 14.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & \frac{4}{3} & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 15

Perform the row operation $R_{1} = R_{1} - \frac{4}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Этап 15.1

Perform the row operation $R_{1} = R_{1} - \frac{4}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{4}{3} \cdot 0 & \frac{25}{18} - \frac{4}{3} \cdot 0 & \frac{4}{3} - \frac{4}{3} \cdot 1 & 0 - \frac{4}{3} \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 15.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{25}{18} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 16

Perform the row operation $R_{1} = R_{1} - \frac{25}{18} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Этап 16.1

Perform the row operation $R_{1} = R_{1} - \frac{25}{18} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{25}{18} \cdot 0 & \frac{25}{18} - \frac{25}{18} \cdot 1 & 0 - \frac{25}{18} \cdot 0 & 0 - \frac{25}{18} \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Этап 16.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$