Đại số tuyến tính Ví dụ

$[\begin{array}{c} 0 & 3 & - 2 & 0 & 6 \\ 2 & 1 & 0 & 3 & 2 \\ 0 & - 1 & 2 & 1 & 0 \\ 1 & 0 & 0 & - 1 & 3 \\ 1 & 3 & - 5 & 8 & 3 \end{array}]$

Bước 1

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 1.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 2 & 1 & 0 & 3 & 2 \\ 0 & 3 & - 2 & 0 & 6 \\ 0 & - 1 & 2 & 1 & 0 \\ 1 & 0 & 0 & - 1 & 3 \\ 1 & 3 & - 5 & 8 & 3 \end{array}]$

Bước 1.2

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

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Bước 1.2.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{1}{2} & \frac{0}{2} & \frac{3}{2} & \frac{2}{2} \\ 0 & 3 & - 2 & 0 & 6 \\ 0 & - 1 & 2 & 1 & 0 \\ 1 & 0 & 0 & - 1 & 3 \\ 1 & 3 & - 5 & 8 & 3 \end{array}]$

Bước 1.2.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 3 & - 2 & 0 & 6 \\ 0 & - 1 & 2 & 1 & 0 \\ 1 & 0 & 0 & - 1 & 3 \\ 1 & 3 & - 5 & 8 & 3 \end{array}]$

Bước 1.3

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

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Bước 1.3.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 3 & - 2 & 0 & 6 \\ 0 & - 1 & 2 & 1 & 0 \\ 1 - 1 & 0 - \frac{1}{2} & 0 - 0 & - 1 - \frac{3}{2} & 3 - 1 \\ 1 & 3 & - 5 & 8 & 3 \end{array}]$

Bước 1.3.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 3 & - 2 & 0 & 6 \\ 0 & - 1 & 2 & 1 & 0 \\ 0 & - \frac{1}{2} & 0 & - \frac{5}{2} & 2 \\ 1 & 3 & - 5 & 8 & 3 \end{array}]$

Bước 1.4

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

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Bước 1.4.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 3 & - 2 & 0 & 6 \\ 0 & - 1 & 2 & 1 & 0 \\ 0 & - \frac{1}{2} & 0 & - \frac{5}{2} & 2 \\ 1 - 1 & 3 - \frac{1}{2} & - 5 - 0 & 8 - \frac{3}{2} & 3 - 1 \end{array}]$

Bước 1.4.2

Rút gọn $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 3 & - 2 & 0 & 6 \\ 0 & - 1 & 2 & 1 & 0 \\ 0 & - \frac{1}{2} & 0 & - \frac{5}{2} & 2 \\ 0 & \frac{5}{2} & - 5 & \frac{13}{2} & 2 \end{array}]$

Bước 1.5

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

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Bước 1.5.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ \frac{0}{3} & \frac{3}{3} & \frac{- 2}{3} & \frac{0}{3} & \frac{6}{3} \\ 0 & - 1 & 2 & 1 & 0 \\ 0 & - \frac{1}{2} & 0 & - \frac{5}{2} & 2 \\ 0 & \frac{5}{2} & - 5 & \frac{13}{2} & 2 \end{array}]$

Bước 1.5.2

Rút gọn $R_{2}$ .

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

Bước 1.6

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

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Bước 1.6.1

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

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

Bước 1.6.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & \frac{4}{3} & 1 & 2 \\ 0 & - \frac{1}{2} & 0 & - \frac{5}{2} & 2 \\ 0 & \frac{5}{2} & - 5 & \frac{13}{2} & 2 \end{array}]$

Bước 1.7

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

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Bước 1.7.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & \frac{4}{3} & 1 & 2 \\ 0 + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & 0 + \frac{1}{2} (- \frac{2}{3}) & - \frac{5}{2} + \frac{1}{2} \cdot 0 & 2 + \frac{1}{2} \cdot 2 \\ 0 & \frac{5}{2} & - 5 & \frac{13}{2} & 2 \end{array}]$

Bước 1.7.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & \frac{4}{3} & 1 & 2 \\ 0 & 0 & - \frac{1}{3} & - \frac{5}{2} & 3 \\ 0 & \frac{5}{2} & - 5 & \frac{13}{2} & 2 \end{array}]$

Bước 1.8

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

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Bước 1.8.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & \frac{4}{3} & 1 & 2 \\ 0 & 0 & - \frac{1}{3} & - \frac{5}{2} & 3 \\ 0 - \frac{5}{2} \cdot 0 & \frac{5}{2} - \frac{5}{2} \cdot 1 & - 5 - \frac{5}{2} (- \frac{2}{3}) & \frac{13}{2} - \frac{5}{2} \cdot 0 & 2 - \frac{5}{2} \cdot 2 \end{array}]$

Bước 1.8.2

Rút gọn $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & \frac{4}{3} & 1 & 2 \\ 0 & 0 & - \frac{1}{3} & - \frac{5}{2} & 3 \\ 0 & 0 & - \frac{10}{3} & \frac{13}{2} & - 3 \end{array}]$

Bước 1.9

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

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Bước 1.9.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ \frac{3}{4} \cdot 0 & \frac{3}{4} \cdot 0 & \frac{3}{4} \cdot \frac{4}{3} & \frac{3}{4} \cdot 1 & \frac{3}{4} \cdot 2 \\ 0 & 0 & - \frac{1}{3} & - \frac{5}{2} & 3 \\ 0 & 0 & - \frac{10}{3} & \frac{13}{2} & - 3 \end{array}]$

Bước 1.9.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & - \frac{1}{3} & - \frac{5}{2} & 3 \\ 0 & 0 & - \frac{10}{3} & \frac{13}{2} & - 3 \end{array}]$

Bước 1.10

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

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Bước 1.10.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 + \frac{1}{3} \cdot 0 & 0 + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & - \frac{5}{2} + \frac{1}{3} \cdot \frac{3}{4} & 3 + \frac{1}{3} \cdot \frac{3}{2} \\ 0 & 0 & - \frac{10}{3} & \frac{13}{2} & - 3 \end{array}]$

Bước 1.10.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & 0 & - \frac{9}{4} & \frac{7}{2} \\ 0 & 0 & - \frac{10}{3} & \frac{13}{2} & - 3 \end{array}]$

Bước 1.11

Perform the row operation $R_{5} = R_{5} + \frac{10}{3} R_{3}$ to make the entry at $5, 3$ a $0$ .

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Bước 1.11.1

Perform the row operation $R_{5} = R_{5} + \frac{10}{3} R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & 0 & - \frac{9}{4} & \frac{7}{2} \\ 0 + \frac{10}{3} \cdot 0 & 0 + \frac{10}{3} \cdot 0 & - \frac{10}{3} + \frac{10}{3} \cdot 1 & \frac{13}{2} + \frac{10}{3} \cdot \frac{3}{4} & - 3 + \frac{10}{3} \cdot \frac{3}{2} \end{array}]$

Bước 1.11.2

Rút gọn $R_{5}$ .

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

Bước 1.12

Multiply each element of $R_{4}$ by $- \frac{4}{9}$ to make the entry at $4, 4$ a $1$ .

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Bước 1.12.1

Multiply each element of $R_{4}$ by $- \frac{4}{9}$ to make the entry at $4, 4$ a $1$ .

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

Bước 1.12.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{14}{9} \\ 0 & 0 & 0 & 9 & 2 \end{array}]$

Bước 1.13

Perform the row operation $R_{5} = R_{5} - 9 R_{4}$ to make the entry at $5, 4$ a $0$ .

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Bước 1.13.1

Perform the row operation $R_{5} = R_{5} - 9 R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{14}{9} \\ 0 - 9 \cdot 0 & 0 - 9 \cdot 0 & 0 - 9 \cdot 0 & 9 - 9 \cdot 1 & 2 - 9 (- \frac{14}{9}) \end{array}]$

Bước 1.13.2

Rút gọn $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{14}{9} \\ 0 & 0 & 0 & 0 & 16 \end{array}]$

Bước 1.14

Multiply each element of $R_{5}$ by $\frac{1}{16}$ to make the entry at $5, 5$ a $1$ .

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Bước 1.14.1

Multiply each element of $R_{5}$ by $\frac{1}{16}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{14}{9} \\ \frac{0}{16} & \frac{0}{16} & \frac{0}{16} & \frac{0}{16} & \frac{16}{16} \end{array}]$

Bước 1.14.2

Rút gọn $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{14}{9} \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.15

Perform the row operation $R_{4} = R_{4} + \frac{14}{9} R_{5}$ to make the entry at $4, 5$ a $0$ .

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Bước 1.15.1

Perform the row operation $R_{4} = R_{4} + \frac{14}{9} R_{5}$ to make the entry at $4, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 & 0 & 1 & \frac{3}{4} & \frac{3}{2} \\ 0 + \frac{14}{9} \cdot 0 & 0 + \frac{14}{9} \cdot 0 & 0 + \frac{14}{9} \cdot 0 & 1 + \frac{14}{9} \cdot 0 & - \frac{14}{9} + \frac{14}{9} \cdot 1 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.15.2

Rút gọn $R_{4}$ .

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

Bước 1.16

Perform the row operation $R_{3} = R_{3} - \frac{3}{2} R_{5}$ to make the entry at $3, 5$ a $0$ .

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Bước 1.16.1

Perform the row operation $R_{3} = R_{3} - \frac{3}{2} R_{5}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 & 1 & - \frac{2}{3} & 0 & 2 \\ 0 - \frac{3}{2} \cdot 0 & 0 - \frac{3}{2} \cdot 0 & 1 - \frac{3}{2} \cdot 0 & \frac{3}{4} - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.16.2

Rút gọn $R_{3}$ .

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

Bước 1.17

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

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Bước 1.17.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 1 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & - \frac{2}{3} - 2 \cdot 0 & 0 - 2 \cdot 0 & 2 - 2 \cdot 1 \\ 0 & 0 & 1 & \frac{3}{4} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.17.2

Rút gọn $R_{2}$ .

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

Bước 1.18

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

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Bước 1.18.1

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

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

Bước 1.18.2

Rút gọn $R_{1}$ .

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

Bước 1.19

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

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Bước 1.19.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & \frac{3}{2} & 0 \\ 0 & 1 & - \frac{2}{3} & 0 & 0 \\ 0 - \frac{3}{4} \cdot 0 & 0 - \frac{3}{4} \cdot 0 & 1 - \frac{3}{4} \cdot 0 & \frac{3}{4} - \frac{3}{4} \cdot 1 & 0 - \frac{3}{4} \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.19.2

Rút gọn $R_{3}$ .

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

Bước 1.20

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{4}$ to make the entry at $1, 4$ a $0$ .

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Bước 1.20.1

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{3}{2} \cdot 0 & \frac{1}{2} - \frac{3}{2} \cdot 0 & 0 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & 0 - \frac{3}{2} \cdot 0 \\ 0 & 1 & - \frac{2}{3} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.20.2

Rút gọn $R_{1}$ .

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

Bước 1.21

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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Bước 1.21.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 0 & 0 \\ 0 + \frac{2}{3} \cdot 0 & 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 0 + \frac{2}{3} \cdot 0 & 0 + \frac{2}{3} \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.21.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Bước 1.22

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

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Bước 1.22.1

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

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

Bước 1.22.2

Rút gọn $R_{1}$ .

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

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

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

Bước 3

The rank is the number of pivot columns.

$5$