Đại số tuyến tính Ví dụ

$[\begin{array}{c} - 42 & 54 & 24 & - 30 & - 18 & 18 & 42 \\ 24 & - 36 & - 42 & 12 & 36 & 30 & - 30 \\ - 30 & 42 & 36 & - 30 & 36 & - 12 & - 48 \\ 18 & - 30 & - 48 & 6 & 42 & 24 & - 48 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 \end{array}]$

Bước 1

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

$[\begin{array}{c} - 42 & 54 & 24 & - 30 & - 18 & 18 & 42 & 0 \\ 24 & - 36 & - 42 & 12 & 36 & 30 & - 30 & 0 \\ - 30 & 42 & 36 & - 30 & 36 & - 12 & - 48 & 0 \\ 18 & - 30 & - 48 & 6 & 42 & 24 & - 48 & 0 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 & 0 \end{array}]$

Bước 2

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 2.1

Multiply each element of $R_{1}$ by $- \frac{1}{42}$ to make the entry at $1, 1$ a $1$ .

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Bước 2.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{42}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{42} \cdot - 42 & - \frac{1}{42} \cdot 54 & - \frac{1}{42} \cdot 24 & - \frac{1}{42} \cdot - 30 & - \frac{1}{42} \cdot - 18 & - \frac{1}{42} \cdot 18 & - \frac{1}{42} \cdot 42 & - \frac{1}{42} \cdot 0 \\ 24 & - 36 & - 42 & 12 & 36 & 30 & - 30 & 0 \\ - 30 & 42 & 36 & - 30 & 36 & - 12 & - 48 & 0 \\ 18 & - 30 & - 48 & 6 & 42 & 24 & - 48 & 0 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 & 0 \end{array}]$

Bước 2.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 24 & - 36 & - 42 & 12 & 36 & 30 & - 30 & 0 \\ - 30 & 42 & 36 & - 30 & 36 & - 12 & - 48 & 0 \\ 18 & - 30 & - 48 & 6 & 42 & 24 & - 48 & 0 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 & 0 \end{array}]$

Bước 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 24 - 24 \cdot 1 & - 36 - 24 (- \frac{9}{7}) & - 42 - 24 (- \frac{4}{7}) & 12 - 24 (\frac{5}{7}) & 36 - 24 (\frac{3}{7}) & 30 - 24 (- \frac{3}{7}) & - 30 - 24 \cdot - 1 & 0 - 24 \cdot 0 \\ - 30 & 42 & 36 & - 30 & 36 & - 12 & - 48 & 0 \\ 18 & - 30 & - 48 & 6 & 42 & 24 & - 48 & 0 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 & 0 \end{array}]$

Bước 2.2.2

Rút gọn $R_{2}$ .

Bước 2.3

Perform the row operation $R_{3} = R_{3} + 30 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 2.3.1

Perform the row operation $R_{3} = R_{3} + 30 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & - \frac{36}{7} & - \frac{198}{7} & - \frac{36}{7} & \frac{180}{7} & \frac{282}{7} & - 6 & 0 \\ - 30 + 30 \cdot 1 & 42 + 30 (- \frac{9}{7}) & 36 + 30 (- \frac{4}{7}) & - 30 + 30 (\frac{5}{7}) & 36 + 30 (\frac{3}{7}) & - 12 + 30 (- \frac{3}{7}) & - 48 + 30 \cdot - 1 & 0 + 30 \cdot 0 \\ 18 & - 30 & - 48 & 6 & 42 & 24 & - 48 & 0 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 & 0 \end{array}]$

Bước 2.3.2

Rút gọn $R_{3}$ .

Bước 2.4

Perform the row operation $R_{4} = R_{4} - 18 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Bước 2.4.1

Perform the row operation $R_{4} = R_{4} - 18 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & - \frac{36}{7} & - \frac{198}{7} & - \frac{36}{7} & \frac{180}{7} & \frac{282}{7} & - 6 & 0 \\ 0 & \frac{24}{7} & \frac{132}{7} & - \frac{60}{7} & \frac{342}{7} & - \frac{174}{7} & - 78 & 0 \\ 18 - 18 \cdot 1 & - 30 - 18 (- \frac{9}{7}) & - 48 - 18 (- \frac{4}{7}) & 6 - 18 (\frac{5}{7}) & 42 - 18 (\frac{3}{7}) & 24 - 18 (- \frac{3}{7}) & - 48 - 18 \cdot - 1 & 0 - 18 \cdot 0 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 & 0 \end{array}]$

Bước 2.4.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & - \frac{36}{7} & - \frac{198}{7} & - \frac{36}{7} & \frac{180}{7} & \frac{282}{7} & - 6 & 0 \\ 0 & \frac{24}{7} & \frac{132}{7} & - \frac{60}{7} & \frac{342}{7} & - \frac{174}{7} & - 78 & 0 \\ 0 & - \frac{48}{7} & - \frac{264}{7} & - \frac{48}{7} & \frac{240}{7} & \frac{222}{7} & - 30 & 0 \\ - 36 & 48 & 30 & - 24 & - 24 & - 54 & - 18 & 0 \end{array}]$

Bước 2.5

Perform the row operation $R_{5} = R_{5} + 36 R_{1}$ to make the entry at $5, 1$ a $0$ .

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Bước 2.5.1

Perform the row operation $R_{5} = R_{5} + 36 R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & - \frac{36}{7} & - \frac{198}{7} & - \frac{36}{7} & \frac{180}{7} & \frac{282}{7} & - 6 & 0 \\ 0 & \frac{24}{7} & \frac{132}{7} & - \frac{60}{7} & \frac{342}{7} & - \frac{174}{7} & - 78 & 0 \\ 0 & - \frac{48}{7} & - \frac{264}{7} & - \frac{48}{7} & \frac{240}{7} & \frac{222}{7} & - 30 & 0 \\ - 36 + 36 \cdot 1 & 48 + 36 (- \frac{9}{7}) & 30 + 36 (- \frac{4}{7}) & - 24 + 36 (\frac{5}{7}) & - 24 + 36 (\frac{3}{7}) & - 54 + 36 (- \frac{3}{7}) & - 18 + 36 \cdot - 1 & 0 + 36 \cdot 0 \end{array}]$

Bước 2.5.2

Rút gọn $R_{5}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & - \frac{36}{7} & - \frac{198}{7} & - \frac{36}{7} & \frac{180}{7} & \frac{282}{7} & - 6 & 0 \\ 0 & \frac{24}{7} & \frac{132}{7} & - \frac{60}{7} & \frac{342}{7} & - \frac{174}{7} & - 78 & 0 \\ 0 & - \frac{48}{7} & - \frac{264}{7} & - \frac{48}{7} & \frac{240}{7} & \frac{222}{7} & - 30 & 0 \\ 0 & \frac{12}{7} & \frac{66}{7} & \frac{12}{7} & - \frac{60}{7} & - \frac{486}{7} & - 54 & 0 \end{array}]$

Bước 2.6

Multiply each element of $R_{2}$ by $- \frac{7}{36}$ to make the entry at $2, 2$ a $1$ .

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Bước 2.6.1

Multiply each element of $R_{2}$ by $- \frac{7}{36}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ - \frac{7}{36} \cdot 0 & - \frac{7}{36} (- \frac{36}{7}) & - \frac{7}{36} (- \frac{198}{7}) & - \frac{7}{36} (- \frac{36}{7}) & - \frac{7}{36} \cdot \frac{180}{7} & - \frac{7}{36} \cdot \frac{282}{7} & - \frac{7}{36} \cdot - 6 & - \frac{7}{36} \cdot 0 \\ 0 & \frac{24}{7} & \frac{132}{7} & - \frac{60}{7} & \frac{342}{7} & - \frac{174}{7} & - 78 & 0 \\ 0 & - \frac{48}{7} & - \frac{264}{7} & - \frac{48}{7} & \frac{240}{7} & \frac{222}{7} & - 30 & 0 \\ 0 & \frac{12}{7} & \frac{66}{7} & \frac{12}{7} & - \frac{60}{7} & - \frac{486}{7} & - 54 & 0 \end{array}]$

Bước 2.6.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 & \frac{24}{7} & \frac{132}{7} & - \frac{60}{7} & \frac{342}{7} & - \frac{174}{7} & - 78 & 0 \\ 0 & - \frac{48}{7} & - \frac{264}{7} & - \frac{48}{7} & \frac{240}{7} & \frac{222}{7} & - 30 & 0 \\ 0 & \frac{12}{7} & \frac{66}{7} & \frac{12}{7} & - \frac{60}{7} & - \frac{486}{7} & - 54 & 0 \end{array}]$

Bước 2.7

Perform the row operation $R_{3} = R_{3} - \frac{24}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 2.7.1

Perform the row operation $R_{3} = R_{3} - \frac{24}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 - \frac{24}{7} \cdot 0 & \frac{24}{7} - \frac{24}{7} \cdot 1 & \frac{132}{7} - \frac{24}{7} \cdot \frac{11}{2} & - \frac{60}{7} - \frac{24}{7} \cdot 1 & \frac{342}{7} - \frac{24}{7} \cdot - 5 & - \frac{174}{7} - \frac{24}{7} (- \frac{47}{6}) & - 78 - \frac{24}{7} \cdot \frac{7}{6} & 0 - \frac{24}{7} \cdot 0 \\ 0 & - \frac{48}{7} & - \frac{264}{7} & - \frac{48}{7} & \frac{240}{7} & \frac{222}{7} & - 30 & 0 \\ 0 & \frac{12}{7} & \frac{66}{7} & \frac{12}{7} & - \frac{60}{7} & - \frac{486}{7} & - 54 & 0 \end{array}]$

Bước 2.7.2

Rút gọn $R_{3}$ .

Bước 2.8

Perform the row operation $R_{4} = R_{4} + \frac{48}{7} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Bước 2.8.1

Perform the row operation $R_{4} = R_{4} + \frac{48}{7} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 & 0 & 0 & - 12 & 66 & 2 & - 82 & 0 \\ 0 + \frac{48}{7} \cdot 0 & - \frac{48}{7} + \frac{48}{7} \cdot 1 & - \frac{264}{7} + \frac{48}{7} \cdot \frac{11}{2} & - \frac{48}{7} + \frac{48}{7} \cdot 1 & \frac{240}{7} + \frac{48}{7} \cdot - 5 & \frac{222}{7} + \frac{48}{7} (- \frac{47}{6}) & - 30 + \frac{48}{7} \cdot \frac{7}{6} & 0 + \frac{48}{7} \cdot 0 \\ 0 & \frac{12}{7} & \frac{66}{7} & \frac{12}{7} & - \frac{60}{7} & - \frac{486}{7} & - 54 & 0 \end{array}]$

Bước 2.8.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 & 0 & 0 & - 12 & 66 & 2 & - 82 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 22 & - 22 & 0 \\ 0 & \frac{12}{7} & \frac{66}{7} & \frac{12}{7} & - \frac{60}{7} & - \frac{486}{7} & - 54 & 0 \end{array}]$

Bước 2.9

Perform the row operation $R_{5} = R_{5} - \frac{12}{7} R_{2}$ to make the entry at $5, 2$ a $0$ .

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Bước 2.9.1

Perform the row operation $R_{5} = R_{5} - \frac{12}{7} R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 & 0 & 0 & - 12 & 66 & 2 & - 82 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 22 & - 22 & 0 \\ 0 - \frac{12}{7} \cdot 0 & \frac{12}{7} - \frac{12}{7} \cdot 1 & \frac{66}{7} - \frac{12}{7} \cdot \frac{11}{2} & \frac{12}{7} - \frac{12}{7} \cdot 1 & - \frac{60}{7} - \frac{12}{7} \cdot - 5 & - \frac{486}{7} - \frac{12}{7} (- \frac{47}{6}) & - 54 - \frac{12}{7} \cdot \frac{7}{6} & 0 - \frac{12}{7} \cdot 0 \end{array}]$

Bước 2.9.2

Rút gọn $R_{5}$ .

Bước 2.10

Multiply each element of $R_{3}$ by $- \frac{1}{12}$ to make the entry at $3, 4$ a $1$ .

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Bước 2.10.1

Multiply each element of $R_{3}$ by $- \frac{1}{12}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ - \frac{1}{12} \cdot 0 & - \frac{1}{12} \cdot 0 & - \frac{1}{12} \cdot 0 & - \frac{1}{12} \cdot - 12 & - \frac{1}{12} \cdot 66 & - \frac{1}{12} \cdot 2 & - \frac{1}{12} \cdot - 82 & - \frac{1}{12} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & - 22 & - 22 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 56 & - 56 & 0 \end{array}]$

Bước 2.10.2

Rút gọn $R_{3}$ .

Bước 2.11

Multiply each element of $R_{4}$ by $- \frac{1}{22}$ to make the entry at $4, 6$ a $1$ .

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Bước 2.11.1

Multiply each element of $R_{4}$ by $- \frac{1}{22}$ to make the entry at $4, 6$ a $1$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & - \frac{1}{6} & \frac{41}{6} & 0 \\ - \frac{1}{22} \cdot 0 & - \frac{1}{22} \cdot 0 & - \frac{1}{22} \cdot 0 & - \frac{1}{22} \cdot 0 & - \frac{1}{22} \cdot 0 & - \frac{1}{22} \cdot - 22 & - \frac{1}{22} \cdot - 22 & - \frac{1}{22} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & - 56 & - 56 & 0 \end{array}]$

Bước 2.11.2

Rút gọn $R_{4}$ .

Bước 2.12

Perform the row operation $R_{5} = R_{5} + 56 R_{4}$ to make the entry at $5, 6$ a $0$ .

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Bước 2.12.1

Perform the row operation $R_{5} = R_{5} + 56 R_{4}$ to make the entry at $5, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & - \frac{1}{6} & \frac{41}{6} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 + 56 \cdot 0 & 0 + 56 \cdot 0 & 0 + 56 \cdot 0 & 0 + 56 \cdot 0 & 0 + 56 \cdot 0 & - 56 + 56 \cdot 1 & - 56 + 56 \cdot 1 & 0 + 56 \cdot 0 \end{array}]$

Bước 2.12.2

Rút gọn $R_{5}$ .

Bước 2.13

Perform the row operation $R_{3} = R_{3} + \frac{1}{6} R_{4}$ to make the entry at $3, 6$ a $0$ .

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Bước 2.13.1

Perform the row operation $R_{3} = R_{3} + \frac{1}{6} R_{4}$ to make the entry at $3, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & - \frac{47}{6} & \frac{7}{6} & 0 \\ 0 + \frac{1}{6} \cdot 0 & 0 + \frac{1}{6} \cdot 0 & 0 + \frac{1}{6} \cdot 0 & 1 + \frac{1}{6} \cdot 0 & - \frac{11}{2} + \frac{1}{6} \cdot 0 & - \frac{1}{6} + \frac{1}{6} \cdot 1 & \frac{41}{6} + \frac{1}{6} \cdot 1 & 0 + \frac{1}{6} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.13.2

Rút gọn $R_{3}$ .

Bước 2.14

Perform the row operation $R_{2} = R_{2} + \frac{47}{6} R_{4}$ to make the entry at $2, 6$ a $0$ .

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Bước 2.14.1

Perform the row operation $R_{2} = R_{2} + \frac{47}{6} R_{4}$ to make the entry at $2, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 + \frac{47}{6} \cdot 0 & 1 + \frac{47}{6} \cdot 0 & \frac{11}{2} + \frac{47}{6} \cdot 0 & 1 + \frac{47}{6} \cdot 0 & - 5 + \frac{47}{6} \cdot 0 & - \frac{47}{6} + \frac{47}{6} \cdot 1 & \frac{7}{6} + \frac{47}{6} \cdot 1 & 0 + \frac{47}{6} \cdot 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.14.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & - \frac{3}{7} & - 1 & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & 0 & 9 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.15

Perform the row operation $R_{1} = R_{1} + \frac{3}{7} R_{4}$ to make the entry at $1, 6$ a $0$ .

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Bước 2.15.1

Perform the row operation $R_{1} = R_{1} + \frac{3}{7} R_{4}$ to make the entry at $1, 6$ a $0$ .

$[\begin{array}{c} 1 + \frac{3}{7} \cdot 0 & - \frac{9}{7} + \frac{3}{7} \cdot 0 & - \frac{4}{7} + \frac{3}{7} \cdot 0 & \frac{5}{7} + \frac{3}{7} \cdot 0 & \frac{3}{7} + \frac{3}{7} \cdot 0 & - \frac{3}{7} + \frac{3}{7} \cdot 1 & - 1 + \frac{3}{7} \cdot 1 & 0 + \frac{3}{7} \cdot 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & 0 & 9 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.15.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & 0 & - \frac{4}{7} & 0 \\ 0 & 1 & \frac{11}{2} & 1 & - 5 & 0 & 9 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.16

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

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Bước 2.16.1

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

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & 0 & - \frac{4}{7} & 0 \\ 0 - 0 & 1 - 0 & \frac{11}{2} - 0 & 1 - 1 & - 5 + \frac{11}{2} & 0 - 0 & 9 - 7 & 0 - 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.16.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & \frac{5}{7} & \frac{3}{7} & 0 & - \frac{4}{7} & 0 \\ 0 & 1 & \frac{11}{2} & 0 & \frac{1}{2} & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.17

Perform the row operation $R_{1} = R_{1} - \frac{5}{7} R_{3}$ to make the entry at $1, 4$ a $0$ .

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Bước 2.17.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{7} R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{7} \cdot 0 & - \frac{9}{7} - \frac{5}{7} \cdot 0 & - \frac{4}{7} - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 & \frac{3}{7} - \frac{5}{7} (- \frac{11}{2}) & 0 - \frac{5}{7} \cdot 0 & - \frac{4}{7} - \frac{5}{7} \cdot 7 & 0 - \frac{5}{7} \cdot 0 \\ 0 & 1 & \frac{11}{2} & 0 & \frac{1}{2} & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.17.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{9}{7} & - \frac{4}{7} & 0 & \frac{61}{14} & 0 & - \frac{39}{7} & 0 \\ 0 & 1 & \frac{11}{2} & 0 & \frac{1}{2} & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.18

Perform the row operation $R_{1} = R_{1} + \frac{9}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Bước 2.18.1

Perform the row operation $R_{1} = R_{1} + \frac{9}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{9}{7} \cdot 0 & - \frac{9}{7} + \frac{9}{7} \cdot 1 & - \frac{4}{7} + \frac{9}{7} \cdot \frac{11}{2} & 0 + \frac{9}{7} \cdot 0 & \frac{61}{14} + \frac{9}{7} \cdot \frac{1}{2} & 0 + \frac{9}{7} \cdot 0 & - \frac{39}{7} + \frac{9}{7} \cdot 2 & 0 + \frac{9}{7} \cdot 0 \\ 0 & 1 & \frac{11}{2} & 0 & \frac{1}{2} & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 2.18.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{13}{2} & 0 & 5 & 0 & - 3 & 0 \\ 0 & 1 & \frac{11}{2} & 0 & \frac{1}{2} & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & - \frac{11}{2} & 0 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 3

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

$x_{1} + \frac{13}{2} x_{3} + 5 x_{5} - 3 x_{7} = 0$

$x_{2} + \frac{11}{2} x_{3} + \frac{1}{2} x_{5} + 2 x_{7} = 0$

$x_{4} - \frac{11}{2} x_{5} + 7 x_{7} = 0$

$x_{6} + x_{7} = 0$

$0 = 0$

Bước 4

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

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \end{array}] = [\begin{array}{c} - \frac{13 x_{3}}{2} - 5 x_{5} + 3 x_{7} \\ - \frac{11 x_{3}}{2} - \frac{x_{5}}{2} - 2 x_{7} \\ x_{3} \\ \frac{11 x_{5}}{2} - 7 x_{7} \\ x_{5} \\ - x_{7} \\ x_{7} \end{array}]$

Bước 5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \end{array}] = x_{3} [\begin{array}{c} - \frac{13}{2} \\ - \frac{11}{2} \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}] + x_{5} [\begin{array}{c} - 5 \\ - \frac{1}{2} \\ 0 \\ \frac{11}{2} \\ 1 \\ 0 \\ 0 \end{array}] + x_{7} [\begin{array}{c} 3 \\ - 2 \\ 0 \\ - 7 \\ 0 \\ - 1 \\ 1 \end{array}]$

Bước 6

Write as a solution set.

${x_{3} [\begin{array}{c} - \frac{13}{2} \\ - \frac{11}{2} \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}] + x_{5} [\begin{array}{c} - 5 \\ - \frac{1}{2} \\ 0 \\ \frac{11}{2} \\ 1 \\ 0 \\ 0 \end{array}] + x_{7} [\begin{array}{c} 3 \\ - 2 \\ 0 \\ - 7 \\ 0 \\ - 1 \\ 1 \end{array}] | x_{3}, x_{5}, x_{7} \in R}$