Линейная алгебра Примеры

$[\begin{array}{c} 63 & - 36 & 45 & - 27 & 54 \\ - 81 & 54 & - 63 & 45 & - 72 \\ - 36 & 63 & - 54 & 72 & - 45 \\ 45 & - 18 & 45 & - 9 & 36 \\ 27 & - 54 & - 54 & - 63 & 36 \\ - 27 & - 45 & 18 & - 36 & 81 \\ - 63 & 45 & 72 & 72 & 27 \end{array}]$

Этап 1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 63 & - 36 & 45 & - 27 & 54 & 0 \\ - 81 & 54 & - 63 & 45 & - 72 & 0 \\ - 36 & 63 & - 54 & 72 & - 45 & 0 \\ 45 & - 18 & 45 & - 9 & 36 & 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2

Приведем матрицу к стандартной форме по строкам.

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Этап 2.1

Multiply each element of $R_{1}$ by $\frac{1}{63}$ to make the entry at $1, 1$ a $1$ .

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Этап 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{63}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{63}{63} & \frac{- 36}{63} & \frac{45}{63} & \frac{- 27}{63} & \frac{54}{63} & \frac{0}{63} \\ - 81 & 54 & - 63 & 45 & - 72 & 0 \\ - 36 & 63 & - 54 & 72 & - 45 & 0 \\ 45 & - 18 & 45 & - 9 & 36 & 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.1.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ - 81 & 54 & - 63 & 45 & - 72 & 0 \\ - 36 & 63 & - 54 & 72 & - 45 & 0 \\ 45 & - 18 & 45 & - 9 & 36 & 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.2

Perform the row operation $R_{2} = R_{2} + 81 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Этап 2.2.1

Perform the row operation $R_{2} = R_{2} + 81 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ - 81 + 81 \cdot 1 & 54 + 81 (- \frac{4}{7}) & - 63 + 81 (\frac{5}{7}) & 45 + 81 (- \frac{3}{7}) & - 72 + 81 (\frac{6}{7}) & 0 + 81 \cdot 0 \\ - 36 & 63 & - 54 & 72 & - 45 & 0 \\ 45 & - 18 & 45 & - 9 & 36 & 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.2.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ - 36 & 63 & - 54 & 72 & - 45 & 0 \\ 45 & - 18 & 45 & - 9 & 36 & 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.3

Perform the row operation $R_{3} = R_{3} + 36 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Этап 2.3.1

Perform the row operation $R_{3} = R_{3} + 36 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ - 36 + 36 \cdot 1 & 63 + 36 (- \frac{4}{7}) & - 54 + 36 (\frac{5}{7}) & 72 + 36 (- \frac{3}{7}) & - 45 + 36 (\frac{6}{7}) & 0 + 36 \cdot 0 \\ 45 & - 18 & 45 & - 9 & 36 & 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.3.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 45 & - 18 & 45 & - 9 & 36 & 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.4

Perform the row operation $R_{4} = R_{4} - 45 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Этап 2.4.1

Perform the row operation $R_{4} = R_{4} - 45 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 45 - 45 \cdot 1 & - 18 - 45 (- \frac{4}{7}) & 45 - 45 (\frac{5}{7}) & - 9 - 45 (- \frac{3}{7}) & 36 - 45 (\frac{6}{7}) & 0 - 45 \cdot 0 \\ 27 & - 54 & - 54 & - 63 & 36 & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.4.2

Упростим $R_{4}$ .

Этап 2.5

Perform the row operation $R_{5} = R_{5} - 27 R_{1}$ to make the entry at $5, 1$ a $0$ .

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Этап 2.5.1

Perform the row operation $R_{5} = R_{5} - 27 R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 27 - 27 \cdot 1 & - 54 - 27 (- \frac{4}{7}) & - 54 - 27 (\frac{5}{7}) & - 63 - 27 (- \frac{3}{7}) & 36 - 27 (\frac{6}{7}) & 0 - 27 \cdot 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.5.2

Упростим $R_{5}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ - 27 & - 45 & 18 & - 36 & 81 & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.6

Perform the row operation $R_{6} = R_{6} + 27 R_{1}$ to make the entry at $6, 1$ a $0$ .

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Этап 2.6.1

Perform the row operation $R_{6} = R_{6} + 27 R_{1}$ to make the entry at $6, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ - 27 + 27 \cdot 1 & - 45 + 27 (- \frac{4}{7}) & 18 + 27 (\frac{5}{7}) & - 36 + 27 (- \frac{3}{7}) & 81 + 27 (\frac{6}{7}) & 0 + 27 \cdot 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.6.2

Упростим $R_{6}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ - 63 & 45 & 72 & 72 & 27 & 0 \end{array}]$

Этап 2.7

Perform the row operation $R_{7} = R_{7} + 63 R_{1}$ to make the entry at $7, 1$ a $0$ .

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Этап 2.7.1

Perform the row operation $R_{7} = R_{7} + 63 R_{1}$ to make the entry at $7, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ - 63 + 63 \cdot 1 & 45 + 63 (- \frac{4}{7}) & 72 + 63 (\frac{5}{7}) & 72 + 63 (- \frac{3}{7}) & 27 + 63 (\frac{6}{7}) & 0 + 63 \cdot 0 \end{array}]$

Этап 2.7.2

Упростим $R_{7}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & \frac{54}{7} & - \frac{36}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.8

Multiply each element of $R_{2}$ by $\frac{7}{54}$ to make the entry at $2, 2$ a $1$ .

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Этап 2.8.1

Multiply each element of $R_{2}$ by $\frac{7}{54}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ \frac{7}{54} \cdot 0 & \frac{7}{54} \cdot \frac{54}{7} & \frac{7}{54} (- \frac{36}{7}) & \frac{7}{54} \cdot \frac{72}{7} & \frac{7}{54} (- \frac{18}{7}) & \frac{7}{54} \cdot 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.8.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & \frac{297}{7} & - \frac{198}{7} & \frac{396}{7} & - \frac{99}{7} & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.9

Perform the row operation $R_{3} = R_{3} - \frac{297}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Этап 2.9.1

Perform the row operation $R_{3} = R_{3} - \frac{297}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 - \frac{297}{7} \cdot 0 & \frac{297}{7} - \frac{297}{7} \cdot 1 & - \frac{198}{7} - \frac{297}{7} (- \frac{2}{3}) & \frac{396}{7} - \frac{297}{7} \cdot \frac{4}{3} & - \frac{99}{7} - \frac{297}{7} (- \frac{1}{3}) & 0 - \frac{297}{7} \cdot 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.9.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{54}{7} & \frac{90}{7} & \frac{72}{7} & - \frac{18}{7} & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.10

Perform the row operation $R_{4} = R_{4} - \frac{54}{7} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Этап 2.10.1

Perform the row operation $R_{4} = R_{4} - \frac{54}{7} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 - \frac{54}{7} \cdot 0 & \frac{54}{7} - \frac{54}{7} \cdot 1 & \frac{90}{7} - \frac{54}{7} (- \frac{2}{3}) & \frac{72}{7} - \frac{54}{7} \cdot \frac{4}{3} & - \frac{18}{7} - \frac{54}{7} (- \frac{1}{3}) & 0 - \frac{54}{7} \cdot 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.10.2

Упростим $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 18 & 0 & 0 & 0 \\ 0 & - \frac{270}{7} & - \frac{513}{7} & - \frac{360}{7} & \frac{90}{7} & 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.11

Perform the row operation $R_{5} = R_{5} + \frac{270}{7} R_{2}$ to make the entry at $5, 2$ a $0$ .

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Этап 2.11.1

Perform the row operation $R_{5} = R_{5} + \frac{270}{7} R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 18 & 0 & 0 & 0 \\ 0 + \frac{270}{7} \cdot 0 & - \frac{270}{7} + \frac{270}{7} \cdot 1 & - \frac{513}{7} + \frac{270}{7} (- \frac{2}{3}) & - \frac{360}{7} + \frac{270}{7} \cdot \frac{4}{3} & \frac{90}{7} + \frac{270}{7} (- \frac{1}{3}) & 0 + \frac{270}{7} \cdot 0 \\ 0 & - \frac{423}{7} & \frac{261}{7} & - \frac{333}{7} & \frac{729}{7} & 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.11.2

Упростим $R_{5}$ .

Этап 2.12

Perform the row operation $R_{6} = R_{6} + \frac{423}{7} R_{2}$ to make the entry at $6, 2$ a $0$ .

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Этап 2.12.1

Perform the row operation $R_{6} = R_{6} + \frac{423}{7} R_{2}$ to make the entry at $6, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 18 & 0 & 0 & 0 \\ 0 & 0 & - 99 & 0 & 0 & 0 \\ 0 + \frac{423}{7} \cdot 0 & - \frac{423}{7} + \frac{423}{7} \cdot 1 & \frac{261}{7} + \frac{423}{7} (- \frac{2}{3}) & - \frac{333}{7} + \frac{423}{7} \cdot \frac{4}{3} & \frac{729}{7} + \frac{423}{7} (- \frac{1}{3}) & 0 + \frac{423}{7} \cdot 0 \\ 0 & 9 & 117 & 45 & 81 & 0 \end{array}]$

Этап 2.12.2

Упростим $R_{6}$ .

Этап 2.13

Perform the row operation $R_{7} = R_{7} - 9 R_{2}$ to make the entry at $7, 2$ a $0$ .

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Этап 2.13.1

Perform the row operation $R_{7} = R_{7} - 9 R_{2}$ to make the entry at $7, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 18 & 0 & 0 & 0 \\ 0 & 0 & - 99 & 0 & 0 & 0 \\ 0 & 0 & - 3 & 33 & 84 & 0 \\ 0 - 9 \cdot 0 & 9 - 9 \cdot 1 & 117 - 9 (- \frac{2}{3}) & 45 - 9 (\frac{4}{3}) & 81 - 9 (- \frac{1}{3}) & 0 - 9 \cdot 0 \end{array}]$

Этап 2.13.2

Упростим $R_{7}$ .

Этап 2.14

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 18 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 99 & 0 & 0 & 0 \\ 0 & 0 & - 3 & 33 & 84 & 0 \\ 0 & 0 & 123 & 33 & 84 & 0 \end{array}]$

Этап 2.15

Multiply each element of $R_{3}$ by $\frac{1}{18}$ to make the entry at $3, 3$ a $1$ .

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Этап 2.15.1

Multiply each element of $R_{3}$ by $\frac{1}{18}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ \frac{0}{18} & \frac{0}{18} & \frac{18}{18} & \frac{0}{18} & \frac{0}{18} & \frac{0}{18} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 99 & 0 & 0 & 0 \\ 0 & 0 & - 3 & 33 & 84 & 0 \\ 0 & 0 & 123 & 33 & 84 & 0 \end{array}]$

Этап 2.15.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 99 & 0 & 0 & 0 \\ 0 & 0 & - 3 & 33 & 84 & 0 \\ 0 & 0 & 123 & 33 & 84 & 0 \end{array}]$

Этап 2.16

Perform the row operation $R_{5} = R_{5} + 99 R_{3}$ to make the entry at $5, 3$ a $0$ .

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Этап 2.16.1

Perform the row operation $R_{5} = R_{5} + 99 R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 + 99 \cdot 0 & 0 + 99 \cdot 0 & - 99 + 99 \cdot 1 & 0 + 99 \cdot 0 & 0 + 99 \cdot 0 & 0 + 99 \cdot 0 \\ 0 & 0 & - 3 & 33 & 84 & 0 \\ 0 & 0 & 123 & 33 & 84 & 0 \end{array}]$

Этап 2.16.2

Упростим $R_{5}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 3 & 33 & 84 & 0 \\ 0 & 0 & 123 & 33 & 84 & 0 \end{array}]$

Этап 2.17

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

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Этап 2.17.1

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

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 + 3 \cdot 0 & 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 33 + 3 \cdot 0 & 84 + 3 \cdot 0 & 0 + 3 \cdot 0 \\ 0 & 0 & 123 & 33 & 84 & 0 \end{array}]$

Этап 2.17.2

Упростим $R_{6}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \\ 0 & 0 & 123 & 33 & 84 & 0 \end{array}]$

Этап 2.18

Perform the row operation $R_{7} = R_{7} - 123 R_{3}$ to make the entry at $7, 3$ a $0$ .

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Этап 2.18.1

Perform the row operation $R_{7} = R_{7} - 123 R_{3}$ to make the entry at $7, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \\ 0 - 123 \cdot 0 & 0 - 123 \cdot 0 & 123 - 123 \cdot 1 & 33 - 123 \cdot 0 & 84 - 123 \cdot 0 & 0 - 123 \cdot 0 \end{array}]$

Этап 2.18.2

Упростим $R_{7}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \end{array}]$

Этап 2.19

Swap $R_{6}$ with $R_{4}$ to put a nonzero entry at $4, 4$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \end{array}]$

Этап 2.20

Multiply each element of $R_{4}$ by $\frac{1}{33}$ to make the entry at $4, 4$ a $1$ .

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Этап 2.20.1

Multiply each element of $R_{4}$ by $\frac{1}{33}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ \frac{0}{33} & \frac{0}{33} & \frac{0}{33} & \frac{33}{33} & \frac{84}{33} & \frac{0}{33} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \end{array}]$

Этап 2.20.2

Упростим $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & \frac{4}{3} & - \frac{1}{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 33 & 84 & 0 \end{array}]$

Этап 2.21

Perform the row operation $R_{7} = R_{7} - 33 R_{4}$ to make the entry at $7, 4$ a $0$ .

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Этап 2.21.1

Perform the row operation $R_{7} = R_{7} - 33 R_{4}$ to make the entry at $7, 4$ a $0$ .

Этап 2.21.2

Упростим $R_{7}$ .

Этап 2.22

Perform the row operation $R_{2} = R_{2} - \frac{4}{3} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Этап 2.22.1

Perform the row operation $R_{2} = R_{2} - \frac{4}{3} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 - \frac{4}{3} \cdot 0 & 1 - \frac{4}{3} \cdot 0 & - \frac{2}{3} - \frac{4}{3} \cdot 0 & \frac{4}{3} - \frac{4}{3} \cdot 1 & - \frac{1}{3} - \frac{4}{3} \cdot \frac{28}{11} & 0 - \frac{4}{3} \cdot 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.22.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & - \frac{3}{7} & \frac{6}{7} & 0 \\ 0 & 1 & - \frac{2}{3} & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.23

Perform the row operation $R_{1} = R_{1} + \frac{3}{7} R_{4}$ to make the entry at $1, 4$ a $0$ .

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Этап 2.23.1

Perform the row operation $R_{1} = R_{1} + \frac{3}{7} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + \frac{3}{7} \cdot 0 & - \frac{4}{7} + \frac{3}{7} \cdot 0 & \frac{5}{7} + \frac{3}{7} \cdot 0 & - \frac{3}{7} + \frac{3}{7} \cdot 1 & \frac{6}{7} + \frac{3}{7} \cdot \frac{28}{11} & 0 + \frac{3}{7} \cdot 0 \\ 0 & 1 & - \frac{2}{3} & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.23.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & 0 & \frac{150}{77} & 0 \\ 0 & 1 & - \frac{2}{3} & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.24

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

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Этап 2.24.1

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

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & 0 & \frac{150}{77} & 0 \\ 0 + \frac{2}{3} \cdot 0 & 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 0 + \frac{2}{3} \cdot 0 & - \frac{41}{11} + \frac{2}{3} \cdot 0 & 0 + \frac{2}{3} \cdot 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.24.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & \frac{5}{7} & 0 & \frac{150}{77} & 0 \\ 0 & 1 & 0 & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.25

Perform the row operation $R_{1} = R_{1} - \frac{5}{7} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Этап 2.25.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{7} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{7} \cdot 0 & - \frac{4}{7} - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 & 0 - \frac{5}{7} \cdot 0 & \frac{150}{77} - \frac{5}{7} \cdot 0 & 0 - \frac{5}{7} \cdot 0 \\ 0 & 1 & 0 & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.25.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{4}{7} & 0 & 0 & \frac{150}{77} & 0 \\ 0 & 1 & 0 & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.26

Perform the row operation $R_{1} = R_{1} + \frac{4}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Этап 2.26.1

Perform the row operation $R_{1} = R_{1} + \frac{4}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{4}{7} \cdot 0 & - \frac{4}{7} + \frac{4}{7} \cdot 1 & 0 + \frac{4}{7} \cdot 0 & 0 + \frac{4}{7} \cdot 0 & \frac{150}{77} + \frac{4}{7} (- \frac{41}{11}) & 0 + \frac{4}{7} \cdot 0 \\ 0 & 1 & 0 & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 2.26.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & - \frac{2}{11} & 0 \\ 0 & 1 & 0 & 0 & - \frac{41}{11} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{28}{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Этап 3

Use the result matrix to declare the final solution to the system of equations.

$x_{1} - \frac{2}{11} x_{5} = 0$

$x_{2} - \frac{41}{11} x_{5} = 0$

$x_{3} = 0$

$x_{4} + \frac{28}{11} x_{5} = 0$

$0 = 0$

Этап 4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{array}] = [\begin{array}{c} \frac{2 x_{5}}{11} \\ \frac{41 x_{5}}{11} \\ 0 \\ - \frac{28 x_{5}}{11} \\ x_{5} \end{array}]$

Этап 5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{array}] = x_{5} [\begin{array}{c} \frac{2}{11} \\ \frac{41}{11} \\ 0 \\ - \frac{28}{11} \\ 1 \end{array}]$

Этап 6

Write as a solution set.

${x_{5} [\begin{array}{c} \frac{2}{11} \\ \frac{41}{11} \\ 0 \\ - \frac{28}{11} \\ 1 \end{array}] | x_{5} \in R}$