Toán hữu hạn Ví dụ

$[\begin{array}{c} 9 & 14 & 1 & 0 & 0 & 0 & 1568 \\ 24 & 10 & 0 & 1 & 0 & 0 & 1448 \\ 51 & 3 & 0 & 0 & 1 & 0 & 1836 \\ - 6 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 1.1

Multiply each element of $R_{1}$ by $\frac{1}{9}$ to make the entry at $1, 1$ a $1$ .

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Bước 1.1.1

Multiply each element of $R_{1}$ by $\frac{1}{9}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{9}{9} & \frac{14}{9} & \frac{1}{9} & \frac{0}{9} & \frac{0}{9} & \frac{0}{9} & \frac{1568}{9} \\ 24 & 10 & 0 & 1 & 0 & 0 & 1448 \\ 51 & 3 & 0 & 0 & 1 & 0 & 1836 \\ - 6 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 24 & 10 & 0 & 1 & 0 & 0 & 1448 \\ 51 & 3 & 0 & 0 & 1 & 0 & 1836 \\ - 6 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.2

Perform the row operation $R_{2} = R_{2} - 24 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 1.2.1

Perform the row operation $R_{2} = R_{2} - 24 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 24 - 24 \cdot 1 & 10 - 24 (\frac{14}{9}) & 0 - 24 (\frac{1}{9}) & 1 - 24 \cdot 0 & 0 - 24 \cdot 0 & 0 - 24 \cdot 0 & 1448 - 24 (\frac{1568}{9}) \\ 51 & 3 & 0 & 0 & 1 & 0 & 1836 \\ - 6 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & - \frac{82}{3} & - \frac{8}{3} & 1 & 0 & 0 & - \frac{8200}{3} \\ 51 & 3 & 0 & 0 & 1 & 0 & 1836 \\ - 6 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.3

Perform the row operation $R_{3} = R_{3} - 51 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 1.3.1

Perform the row operation $R_{3} = R_{3} - 51 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & - \frac{82}{3} & - \frac{8}{3} & 1 & 0 & 0 & - \frac{8200}{3} \\ 51 - 51 \cdot 1 & 3 - 51 (\frac{14}{9}) & 0 - 51 (\frac{1}{9}) & 0 - 51 \cdot 0 & 1 - 51 \cdot 0 & 0 - 51 \cdot 0 & 1836 - 51 (\frac{1568}{9}) \\ - 6 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & - \frac{82}{3} & - \frac{8}{3} & 1 & 0 & 0 & - \frac{8200}{3} \\ 0 & - \frac{229}{3} & - \frac{17}{3} & 0 & 1 & 0 & - \frac{21148}{3} \\ - 6 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.4

Perform the row operation $R_{4} = R_{4} + 6 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Bước 1.4.1

Perform the row operation $R_{4} = R_{4} + 6 R_{1}$ to make the entry at $4, 1$ a $0$ .

Bước 1.4.2

Rút gọn $R_{4}$ .

Bước 1.5

Multiply each element of $R_{2}$ by $- \frac{3}{82}$ to make the entry at $2, 2$ a $1$ .

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Bước 1.5.1

Multiply each element of $R_{2}$ by $- \frac{3}{82}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ - \frac{3}{82} \cdot 0 & - \frac{3}{82} (- \frac{82}{3}) & - \frac{3}{82} (- \frac{8}{3}) & - \frac{3}{82} \cdot 1 & - \frac{3}{82} \cdot 0 & - \frac{3}{82} \cdot 0 & - \frac{3}{82} (- \frac{8200}{3}) \\ 0 & - \frac{229}{3} & - \frac{17}{3} & 0 & 1 & 0 & - \frac{21148}{3} \\ 0 & \frac{25}{3} & \frac{2}{3} & 0 & 0 & 1 & \frac{3136}{3} \end{array}]$

Bước 1.5.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ 0 & - \frac{229}{3} & - \frac{17}{3} & 0 & 1 & 0 & - \frac{21148}{3} \\ 0 & \frac{25}{3} & \frac{2}{3} & 0 & 0 & 1 & \frac{3136}{3} \end{array}]$

Bước 1.6

Perform the row operation $R_{3} = R_{3} + \frac{229}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 1.6.1

Perform the row operation $R_{3} = R_{3} + \frac{229}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ 0 + \frac{229}{3} \cdot 0 & - \frac{229}{3} + \frac{229}{3} \cdot 1 & - \frac{17}{3} + \frac{229}{3} \cdot \frac{4}{41} & 0 + \frac{229}{3} (- \frac{3}{82}) & 1 + \frac{229}{3} \cdot 0 & 0 + \frac{229}{3} \cdot 0 & - \frac{21148}{3} + \frac{229}{3} \cdot 100 \\ 0 & \frac{25}{3} & \frac{2}{3} & 0 & 0 & 1 & \frac{3136}{3} \end{array}]$

Bước 1.6.2

Rút gọn $R_{3}$ .

Bước 1.7

Perform the row operation $R_{4} = R_{4} - \frac{25}{3} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Bước 1.7.1

Perform the row operation $R_{4} = R_{4} - \frac{25}{3} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ 0 & 0 & \frac{73}{41} & - \frac{229}{82} & 1 & 0 & 584 \\ 0 - \frac{25}{3} \cdot 0 & \frac{25}{3} - \frac{25}{3} \cdot 1 & \frac{2}{3} - \frac{25}{3} \cdot \frac{4}{41} & 0 - \frac{25}{3} (- \frac{3}{82}) & 0 - \frac{25}{3} \cdot 0 & 1 - \frac{25}{3} \cdot 0 & \frac{3136}{3} - \frac{25}{3} \cdot 100 \end{array}]$

Bước 1.7.2

Rút gọn $R_{4}$ .

Bước 1.8

Multiply each element of $R_{3}$ by $\frac{41}{73}$ to make the entry at $3, 3$ a $1$ .

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Bước 1.8.1

Multiply each element of $R_{3}$ by $\frac{41}{73}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ \frac{41}{73} \cdot 0 & \frac{41}{73} \cdot 0 & \frac{41}{73} \cdot \frac{73}{41} & \frac{41}{73} (- \frac{229}{82}) & \frac{41}{73} \cdot 1 & \frac{41}{73} \cdot 0 & \frac{41}{73} \cdot 584 \\ 0 & 0 & - \frac{6}{41} & \frac{25}{82} & 0 & 1 & 212 \end{array}]$

Bước 1.8.2

Rút gọn $R_{3}$ .

Bước 1.9

Perform the row operation $R_{4} = R_{4} + \frac{6}{41} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Bước 1.9.1

Perform the row operation $R_{4} = R_{4} + \frac{6}{41} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ 0 & 0 & 1 & - \frac{229}{146} & \frac{41}{73} & 0 & 328 \\ 0 + \frac{6}{41} \cdot 0 & 0 + \frac{6}{41} \cdot 0 & - \frac{6}{41} + \frac{6}{41} \cdot 1 & \frac{25}{82} + \frac{6}{41} (- \frac{229}{146}) & 0 + \frac{6}{41} \cdot \frac{41}{73} & 1 + \frac{6}{41} \cdot 0 & 212 + \frac{6}{41} \cdot 328 \end{array}]$

Bước 1.9.2

Rút gọn $R_{4}$ .

Bước 1.10

Multiply each element of $R_{4}$ by $\frac{146}{11}$ to make the entry at $4, 4$ a $1$ .

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Bước 1.10.1

Multiply each element of $R_{4}$ by $\frac{146}{11}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ 0 & 0 & 1 & - \frac{229}{146} & \frac{41}{73} & 0 & 328 \\ \frac{146}{11} \cdot 0 & \frac{146}{11} \cdot 0 & \frac{146}{11} \cdot 0 & \frac{146}{11} \cdot \frac{11}{146} & \frac{146}{11} \cdot \frac{6}{73} & \frac{146}{11} \cdot 1 & \frac{146}{11} \cdot 260 \end{array}]$

Bước 1.10.2

Rút gọn $R_{4}$ .

Bước 1.11

Perform the row operation $R_{3} = R_{3} + \frac{229}{146} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Bước 1.11.1

Perform the row operation $R_{3} = R_{3} + \frac{229}{146} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ 0 + \frac{229}{146} \cdot 0 & 0 + \frac{229}{146} \cdot 0 & 1 + \frac{229}{146} \cdot 0 & - \frac{229}{146} + \frac{229}{146} \cdot 1 & \frac{41}{73} + \frac{229}{146} \cdot \frac{12}{11} & 0 + \frac{229}{146} \cdot \frac{146}{11} & 328 + \frac{229}{146} \cdot \frac{37960}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.11.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & - \frac{3}{82} & 0 & 0 & 100 \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.12

Perform the row operation $R_{2} = R_{2} + \frac{3}{82} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Bước 1.12.1

Perform the row operation $R_{2} = R_{2} + \frac{3}{82} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 + \frac{3}{82} \cdot 0 & 1 + \frac{3}{82} \cdot 0 & \frac{4}{41} + \frac{3}{82} \cdot 0 & - \frac{3}{82} + \frac{3}{82} \cdot 1 & 0 + \frac{3}{82} \cdot \frac{12}{11} & 0 + \frac{3}{82} \cdot \frac{146}{11} & 100 + \frac{3}{82} \cdot \frac{37960}{11} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.12.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & \frac{4}{41} & 0 & \frac{18}{451} & \frac{219}{451} & \frac{102040}{451} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.13

Perform the row operation $R_{2} = R_{2} - \frac{4}{41} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 1.13.1

Perform the row operation $R_{2} = R_{2} - \frac{4}{41} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 - \frac{4}{41} \cdot 0 & 1 - \frac{4}{41} \cdot 0 & \frac{4}{41} - \frac{4}{41} \cdot 1 & 0 - \frac{4}{41} \cdot 0 & \frac{18}{451} - \frac{4}{41} \cdot \frac{25}{11} & \frac{219}{451} - \frac{4}{41} \cdot \frac{229}{11} & \frac{102040}{451} - \frac{4}{41} \cdot \frac{63148}{11} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.13.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & \frac{1}{9} & 0 & 0 & 0 & \frac{1568}{9} \\ 0 & 1 & 0 & 0 & - \frac{2}{11} & - \frac{17}{11} & - \frac{3672}{11} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.14

Perform the row operation $R_{1} = R_{1} - \frac{1}{9} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Bước 1.14.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{9} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{9} \cdot 0 & \frac{14}{9} - \frac{1}{9} \cdot 0 & \frac{1}{9} - \frac{1}{9} \cdot 1 & 0 - \frac{1}{9} \cdot 0 & 0 - \frac{1}{9} \cdot \frac{25}{11} & 0 - \frac{1}{9} \cdot \frac{229}{11} & \frac{1568}{9} - \frac{1}{9} \cdot \frac{63148}{11} \\ 0 & 1 & 0 & 0 & - \frac{2}{11} & - \frac{17}{11} & - \frac{3672}{11} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.14.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{14}{9} & 0 & 0 & - \frac{25}{99} & - \frac{229}{99} & - \frac{5100}{11} \\ 0 & 1 & 0 & 0 & - \frac{2}{11} & - \frac{17}{11} & - \frac{3672}{11} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.15

Perform the row operation $R_{1} = R_{1} - \frac{14}{9} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Bước 1.15.1

Perform the row operation $R_{1} = R_{1} - \frac{14}{9} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{14}{9} \cdot 0 & \frac{14}{9} - \frac{14}{9} \cdot 1 & 0 - \frac{14}{9} \cdot 0 & 0 - \frac{14}{9} \cdot 0 & - \frac{25}{99} - \frac{14}{9} (- \frac{2}{11}) & - \frac{229}{99} - \frac{14}{9} (- \frac{17}{11}) & - \frac{5100}{11} - \frac{14}{9} (- \frac{3672}{11}) \\ 0 & 1 & 0 & 0 & - \frac{2}{11} & - \frac{17}{11} & - \frac{3672}{11} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 1.15.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & \frac{1}{33} & \frac{1}{11} & \frac{612}{11} \\ 0 & 1 & 0 & 0 & - \frac{2}{11} & - \frac{17}{11} & - \frac{3672}{11} \\ 0 & 0 & 1 & 0 & \frac{25}{11} & \frac{229}{11} & \frac{63148}{11} \\ 0 & 0 & 0 & 1 & \frac{12}{11} & \frac{146}{11} & \frac{37960}{11} \end{array}]$

Bước 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: $a_{11}, a_{22}, a_{33},$ and $a_{44}$

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