Конечная математика Примеры

$[\begin{array}{c} 500 & 50 & 3000 & 1 & 0 & 0 & 0 & 0 & 9000 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 40 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ - 5000 & - 3000 & - 12000 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Этап 1

Приведем матрицу к стандартной форме по строкам.

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Этап 1.1

Multiply each element of $R_{1}$ by $\frac{1}{500}$ to make the entry at $1, 1$ a $1$ .

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Этап 1.1.1

Multiply each element of $R_{1}$ by $\frac{1}{500}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{500}{500} & \frac{50}{500} & \frac{3000}{500} & \frac{1}{500} & \frac{0}{500} & \frac{0}{500} & \frac{0}{500} & \frac{0}{500} & \frac{9000}{500} \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 40 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ - 5000 & - 3000 & - 12000 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Этап 1.1.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 40 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ - 5000 & - 3000 & - 12000 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Этап 1.2

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

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Этап 1.2.1

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

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 1 - 1 & 0 - \frac{1}{10} & 0 - 6 & 0 - \frac{1}{500} & 1 - 0 & 0 - 0 & 0 - 0 & 0 - 0 & 40 - 18 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ - 5000 & - 3000 & - 12000 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Этап 1.2.2

Упростим $R_{2}$ .

Этап 1.3

Perform the row operation $R_{5} = R_{5} + 5000 R_{1}$ to make the entry at $5, 1$ a $0$ .

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Этап 1.3.1

Perform the row operation $R_{5} = R_{5} + 5000 R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & - \frac{1}{10} & - 6 & - \frac{1}{500} & 1 & 0 & 0 & 0 & 22 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ - 5000 + 5000 \cdot 1 & - 3000 + 5000 (\frac{1}{10}) & - 12000 + 5000 \cdot 6 & 0 + 5000 (\frac{1}{500}) & 0 + 5000 \cdot 0 & 0 + 5000 \cdot 0 & 0 + 5000 \cdot 0 & 1 + 5000 \cdot 0 & 0 + 5000 \cdot 18 \end{array}]$

Этап 1.3.2

Упростим $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & - \frac{1}{10} & - 6 & - \frac{1}{500} & 1 & 0 & 0 & 0 & 22 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & - 2500 & 18000 & 10 & 0 & 0 & 0 & 1 & 90000 \end{array}]$

Этап 1.4

Multiply each element of $R_{2}$ by $- 10$ to make the entry at $2, 2$ a $1$ .

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Этап 1.4.1

Multiply each element of $R_{2}$ by $- 10$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ - 10 \cdot 0 & - 10 (- \frac{1}{10}) & - 10 \cdot - 6 & - 10 (- \frac{1}{500}) & - 10 \cdot 1 & - 10 \cdot 0 & - 10 \cdot 0 & - 10 \cdot 0 & - 10 \cdot 22 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & - 2500 & 18000 & 10 & 0 & 0 & 0 & 1 & 90000 \end{array}]$

Этап 1.4.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & - 2500 & 18000 & 10 & 0 & 0 & 0 & 1 & 90000 \end{array}]$

Этап 1.5

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

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Этап 1.5.1

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

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 - 0 & 1 - 1 & 0 - 60 & 0 - \frac{1}{50} & 0 + 10 & 1 - 0 & 0 - 0 & 0 - 0 & 60 + 220 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & - 2500 & 18000 & 10 & 0 & 0 & 0 & 1 & 90000 \end{array}]$

Этап 1.5.2

Упростим $R_{3}$ .

Этап 1.6

Perform the row operation $R_{5} = R_{5} + 2500 R_{2}$ to make the entry at $5, 2$ a $0$ .

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Этап 1.6.1

Perform the row operation $R_{5} = R_{5} + 2500 R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & - 60 & - \frac{1}{50} & 10 & 1 & 0 & 0 & 280 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 + 2500 \cdot 0 & - 2500 + 2500 \cdot 1 & 18000 + 2500 \cdot 60 & 10 + 2500 (\frac{1}{50}) & 0 + 2500 \cdot - 10 & 0 + 2500 \cdot 0 & 0 + 2500 \cdot 0 & 1 + 2500 \cdot 0 & 90000 + 2500 \cdot - 220 \end{array}]$

Этап 1.6.2

Упростим $R_{5}$ .

Этап 1.7

Multiply each element of $R_{3}$ by $- \frac{1}{60}$ to make the entry at $3, 3$ a $1$ .

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Этап 1.7.1

Multiply each element of $R_{3}$ by $- \frac{1}{60}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ - \frac{1}{60} \cdot 0 & - \frac{1}{60} \cdot 0 & - \frac{1}{60} \cdot - 60 & - \frac{1}{60} (- \frac{1}{50}) & - \frac{1}{60} \cdot 10 & - \frac{1}{60} \cdot 1 & - \frac{1}{60} \cdot 0 & - \frac{1}{60} \cdot 0 & - \frac{1}{60} \cdot 280 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 168000 & 60 & - 25000 & 0 & 0 & 1 & - 460000 \end{array}]$

Этап 1.7.2

Упростим $R_{3}$ .

Этап 1.8

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

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Этап 1.8.1

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

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & - \frac{1}{6} & - \frac{1}{60} & 0 & 0 & - \frac{14}{3} \\ 0 - 0 & 0 - 0 & 1 - 1 & 0 - \frac{1}{3000} & 0 + \frac{1}{6} & 0 + \frac{1}{60} & 1 - 0 & 0 - 0 & 20 + \frac{14}{3} \\ 0 & 0 & 168000 & 60 & - 25000 & 0 & 0 & 1 & - 460000 \end{array}]$

Этап 1.8.2

Упростим $R_{4}$ .

Этап 1.9

Perform the row operation $R_{5} = R_{5} - 168000 R_{3}$ to make the entry at $5, 3$ a $0$ .

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Этап 1.9.1

Perform the row operation $R_{5} = R_{5} - 168000 R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & - \frac{1}{6} & - \frac{1}{60} & 0 & 0 & - \frac{14}{3} \\ 0 & 0 & 0 & - \frac{1}{3000} & \frac{1}{6} & \frac{1}{60} & 1 & 0 & \frac{74}{3} \\ 0 - 168000 \cdot 0 & 0 - 168000 \cdot 0 & 168000 - 168000 \cdot 1 & 60 - 168000 (\frac{1}{3000}) & - 25000 - 168000 (- \frac{1}{6}) & 0 - 168000 (- \frac{1}{60}) & 0 - 168000 \cdot 0 & 1 - 168000 \cdot 0 & - 460000 - 168000 (- \frac{14}{3}) \end{array}]$

Этап 1.9.2

Упростим $R_{5}$ .

Этап 1.10

Multiply each element of $R_{4}$ by $- 3000$ to make the entry at $4, 4$ a $1$ .

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Этап 1.10.1

Multiply each element of $R_{4}$ by $- 3000$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & - \frac{1}{6} & - \frac{1}{60} & 0 & 0 & - \frac{14}{3} \\ - 3000 \cdot 0 & - 3000 \cdot 0 & - 3000 \cdot 0 & - 3000 (- \frac{1}{3000}) & - 3000 (\frac{1}{6}) & - 3000 (\frac{1}{60}) & - 3000 \cdot 1 & - 3000 \cdot 0 & - 3000 (\frac{74}{3}) \\ 0 & 0 & 0 & 4 & 3000 & 2800 & 0 & 1 & 324000 \end{array}]$

Этап 1.10.2

Упростим $R_{4}$ .

Этап 1.11

Perform the row operation $R_{5} = R_{5} - 4 R_{4}$ to make the entry at $5, 4$ a $0$ .

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Этап 1.11.1

Perform the row operation $R_{5} = R_{5} - 4 R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & - \frac{1}{6} & - \frac{1}{60} & 0 & 0 & - \frac{14}{3} \\ 0 & 0 & 0 & 1 & - 500 & - 50 & - 3000 & 0 & - 74000 \\ 0 - 4 \cdot 0 & 0 - 4 \cdot 0 & 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & 3000 - 4 \cdot - 500 & 2800 - 4 \cdot - 50 & 0 - 4 \cdot - 3000 & 1 - 4 \cdot 0 & 324000 - 4 \cdot - 74000 \end{array}]$

Этап 1.11.2

Упростим $R_{5}$ .

Этап 1.12

Multiply each element of $R_{5}$ by $\frac{1}{5000}$ to make the entry at $5, 5$ a $1$ .

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Этап 1.12.1

Multiply each element of $R_{5}$ by $\frac{1}{5000}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & - \frac{1}{6} & - \frac{1}{60} & 0 & 0 & - \frac{14}{3} \\ 0 & 0 & 0 & 1 & - 500 & - 50 & - 3000 & 0 & - 74000 \\ \frac{0}{5000} & \frac{0}{5000} & \frac{0}{5000} & \frac{0}{5000} & \frac{5000}{5000} & \frac{3000}{5000} & \frac{12000}{5000} & \frac{1}{5000} & \frac{620000}{5000} \end{array}]$

Этап 1.12.2

Упростим $R_{5}$ .

Этап 1.13

Perform the row operation $R_{4} = R_{4} + 500 R_{5}$ to make the entry at $4, 5$ a $0$ .

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Этап 1.13.1

Perform the row operation $R_{4} = R_{4} + 500 R_{5}$ to make the entry at $4, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & - \frac{1}{6} & - \frac{1}{60} & 0 & 0 & - \frac{14}{3} \\ 0 + 500 \cdot 0 & 0 + 500 \cdot 0 & 0 + 500 \cdot 0 & 1 + 500 \cdot 0 & - 500 + 500 \cdot 1 & - 50 + 500 (\frac{3}{5}) & - 3000 + 500 (\frac{12}{5}) & 0 + 500 (\frac{1}{5000}) & - 74000 + 500 \cdot 124 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.13.2

Упростим $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & - \frac{1}{6} & - \frac{1}{60} & 0 & 0 & - \frac{14}{3} \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.14

Perform the row operation $R_{3} = R_{3} + \frac{1}{6} R_{5}$ to make the entry at $3, 5$ a $0$ .

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Этап 1.14.1

Perform the row operation $R_{3} = R_{3} + \frac{1}{6} R_{5}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 + \frac{1}{6} \cdot 0 & 0 + \frac{1}{6} \cdot 0 & 1 + \frac{1}{6} \cdot 0 & \frac{1}{3000} + \frac{1}{6} \cdot 0 & - \frac{1}{6} + \frac{1}{6} \cdot 1 & - \frac{1}{60} + \frac{1}{6} \cdot \frac{3}{5} & 0 + \frac{1}{6} \cdot \frac{12}{5} & 0 + \frac{1}{6} \cdot \frac{1}{5000} & - \frac{14}{3} + \frac{1}{6} \cdot 124 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.14.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & - 10 & 0 & 0 & 0 & - 220 \\ 0 & 0 & 1 & \frac{1}{3000} & 0 & \frac{1}{12} & \frac{2}{5} & \frac{1}{30000} & 16 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.15

Perform the row operation $R_{2} = R_{2} + 10 R_{5}$ to make the entry at $2, 5$ a $0$ .

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Этап 1.15.1

Perform the row operation $R_{2} = R_{2} + 10 R_{5}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 + 10 \cdot 0 & 1 + 10 \cdot 0 & 60 + 10 \cdot 0 & \frac{1}{50} + 10 \cdot 0 & - 10 + 10 \cdot 1 & 0 + 10 (\frac{3}{5}) & 0 + 10 (\frac{12}{5}) & 0 + 10 (\frac{1}{5000}) & - 220 + 10 \cdot 124 \\ 0 & 0 & 1 & \frac{1}{3000} & 0 & \frac{1}{12} & \frac{2}{5} & \frac{1}{30000} & 16 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.15.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & 0 & 6 & 24 & \frac{1}{500} & 1020 \\ 0 & 0 & 1 & \frac{1}{3000} & 0 & \frac{1}{12} & \frac{2}{5} & \frac{1}{30000} & 16 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.16

Perform the row operation $R_{3} = R_{3} - \frac{1}{3000} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Этап 1.16.1

Perform the row operation $R_{3} = R_{3} - \frac{1}{3000} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & 0 & 6 & 24 & \frac{1}{500} & 1020 \\ 0 - \frac{1}{3000} \cdot 0 & 0 - \frac{1}{3000} \cdot 0 & 1 - \frac{1}{3000} \cdot 0 & \frac{1}{3000} - \frac{1}{3000} \cdot 1 & 0 - \frac{1}{3000} \cdot 0 & \frac{1}{12} - \frac{1}{3000} \cdot 250 & \frac{2}{5} - \frac{1}{3000} \cdot - 1800 & \frac{1}{30000} - \frac{1}{3000} \cdot \frac{1}{10} & 16 - \frac{1}{3000} \cdot - 12000 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.16.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & \frac{1}{50} & 0 & 6 & 24 & \frac{1}{500} & 1020 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.17

Perform the row operation $R_{2} = R_{2} - \frac{1}{50} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Этап 1.17.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{50} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 - \frac{1}{50} \cdot 0 & 1 - \frac{1}{50} \cdot 0 & 60 - \frac{1}{50} \cdot 0 & \frac{1}{50} - \frac{1}{50} \cdot 1 & 0 - \frac{1}{50} \cdot 0 & 6 - \frac{1}{50} \cdot 250 & 24 - \frac{1}{50} \cdot - 1800 & \frac{1}{500} - \frac{1}{50} \cdot \frac{1}{10} & 1020 - \frac{1}{50} \cdot - 12000 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.17.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & \frac{1}{500} & 0 & 0 & 0 & 0 & 18 \\ 0 & 1 & 60 & 0 & 0 & 1 & 60 & 0 & 1260 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.18

Perform the row operation $R_{1} = R_{1} - \frac{1}{500} R_{4}$ to make the entry at $1, 4$ a $0$ .

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Этап 1.18.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{500} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{500} \cdot 0 & \frac{1}{10} - \frac{1}{500} \cdot 0 & 6 - \frac{1}{500} \cdot 0 & \frac{1}{500} - \frac{1}{500} \cdot 1 & 0 - \frac{1}{500} \cdot 0 & 0 - \frac{1}{500} \cdot 250 & 0 - \frac{1}{500} \cdot - 1800 & 0 - \frac{1}{500} \cdot \frac{1}{10} & 18 - \frac{1}{500} \cdot - 12000 \\ 0 & 1 & 60 & 0 & 0 & 1 & 60 & 0 & 1260 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.18.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & 0 & 0 & - \frac{1}{2} & \frac{18}{5} & - \frac{1}{5000} & 42 \\ 0 & 1 & 60 & 0 & 0 & 1 & 60 & 0 & 1260 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.19

Perform the row operation $R_{2} = R_{2} - 60 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Этап 1.19.1

Perform the row operation $R_{2} = R_{2} - 60 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & 0 & 0 & - \frac{1}{2} & \frac{18}{5} & - \frac{1}{5000} & 42 \\ 0 - 60 \cdot 0 & 1 - 60 \cdot 0 & 60 - 60 \cdot 1 & 0 - 60 \cdot 0 & 0 - 60 \cdot 0 & 1 - 60 \cdot 0 & 60 - 60 \cdot 1 & 0 - 60 \cdot 0 & 1260 - 60 \cdot 20 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.19.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 6 & 0 & 0 & - \frac{1}{2} & \frac{18}{5} & - \frac{1}{5000} & 42 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.20

Perform the row operation $R_{1} = R_{1} - 6 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Этап 1.20.1

Perform the row operation $R_{1} = R_{1} - 6 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 6 \cdot 0 & \frac{1}{10} - 6 \cdot 0 & 6 - 6 \cdot 1 & 0 - 6 \cdot 0 & 0 - 6 \cdot 0 & - \frac{1}{2} - 6 \cdot 0 & \frac{18}{5} - 6 \cdot 1 & - \frac{1}{5000} - 6 \cdot 0 & 42 - 6 \cdot 20 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.20.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{10} & 0 & 0 & 0 & - \frac{1}{2} & - \frac{12}{5} & - \frac{1}{5000} & - 78 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.21

Perform the row operation $R_{1} = R_{1} - \frac{1}{10} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Этап 1.21.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{10} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{10} \cdot 0 & \frac{1}{10} - \frac{1}{10} \cdot 1 & 0 - \frac{1}{10} \cdot 0 & 0 - \frac{1}{10} \cdot 0 & 0 - \frac{1}{10} \cdot 0 & - \frac{1}{2} - \frac{1}{10} \cdot 1 & - \frac{12}{5} - \frac{1}{10} \cdot 0 & - \frac{1}{5000} - \frac{1}{10} \cdot 0 & - 78 - \frac{1}{10} \cdot 60 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 1.21.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & - \frac{3}{5} & - \frac{12}{5} & - \frac{1}{5000} & - 84 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 60 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 20 \\ 0 & 0 & 0 & 1 & 0 & 250 & - 1800 & \frac{1}{10} & - 12000 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{5} & \frac{12}{5} & \frac{1}{5000} & 124 \end{array}]$

Этап 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22}, a_{33}, a_{44},$ and $a_{55}$

Pivot Columns: $1, 2, 3, 4,$ and $5$