Đại số tuyến tính Ví dụ

$[\begin{array}{c} 6 & 0 & 1 & 0 & 0 \\ 3 & 1 & 11 & 0 & 0 \\ 5 & - 2 & 14 & 0 & 0 \\ 4 & - 5 & 14 & 5 & 0 \end{array}]$

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 $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Bước 1.1.1

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

$[\begin{array}{c} \frac{6}{6} & \frac{0}{6} & \frac{1}{6} & \frac{0}{6} & \frac{0}{6} \\ 3 & 1 & 11 & 0 & 0 \\ 5 & - 2 & 14 & 0 & 0 \\ 4 & - 5 & 14 & 5 & 0 \end{array}]$

Bước 1.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 3 & 1 & 11 & 0 & 0 \\ 5 & - 2 & 14 & 0 & 0 \\ 4 & - 5 & 14 & 5 & 0 \end{array}]$

Bước 1.2

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

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Bước 1.2.1

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

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 3 - 3 \cdot 1 & 1 - 3 \cdot 0 & 11 - 3 (\frac{1}{6}) & 0 - 3 \cdot 0 & 0 - 3 \cdot 0 \\ 5 & - 2 & 14 & 0 & 0 \\ 4 & - 5 & 14 & 5 & 0 \end{array}]$

Bước 1.2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 5 & - 2 & 14 & 0 & 0 \\ 4 & - 5 & 14 & 5 & 0 \end{array}]$

Bước 1.3

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

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Bước 1.3.1

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

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 5 - 5 \cdot 1 & - 2 - 5 \cdot 0 & 14 - 5 (\frac{1}{6}) & 0 - 5 \cdot 0 & 0 - 5 \cdot 0 \\ 4 & - 5 & 14 & 5 & 0 \end{array}]$

Bước 1.3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & - 2 & \frac{79}{6} & 0 & 0 \\ 4 & - 5 & 14 & 5 & 0 \end{array}]$

Bước 1.4

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

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Bước 1.4.1

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

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & - 2 & \frac{79}{6} & 0 & 0 \\ 4 - 4 \cdot 1 & - 5 - 4 \cdot 0 & 14 - 4 (\frac{1}{6}) & 5 - 4 \cdot 0 & 0 - 4 \cdot 0 \end{array}]$

Bước 1.4.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & - 2 & \frac{79}{6} & 0 & 0 \\ 0 & - 5 & \frac{40}{3} & 5 & 0 \end{array}]$

Bước 1.5

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

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Bước 1.5.1

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

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & \frac{79}{6} + 2 (\frac{21}{2}) & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 \\ 0 & - 5 & \frac{40}{3} & 5 & 0 \end{array}]$

Bước 1.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & \frac{205}{6} & 0 & 0 \\ 0 & - 5 & \frac{40}{3} & 5 & 0 \end{array}]$

Bước 1.6

Perform the row operation $R_{4} = R_{4} + 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Bước 1.6.1

Perform the row operation $R_{4} = R_{4} + 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & \frac{205}{6} & 0 & 0 \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & \frac{40}{3} + 5 (\frac{21}{2}) & 5 + 5 \cdot 0 & 0 + 5 \cdot 0 \end{array}]$

Bước 1.6.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & \frac{205}{6} & 0 & 0 \\ 0 & 0 & \frac{395}{6} & 5 & 0 \end{array}]$

Bước 1.7

Multiply each element of $R_{3}$ by $\frac{6}{205}$ to make the entry at $3, 3$ a $1$ .

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Bước 1.7.1

Multiply each element of $R_{3}$ by $\frac{6}{205}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ \frac{6}{205} \cdot 0 & \frac{6}{205} \cdot 0 & \frac{6}{205} \cdot \frac{205}{6} & \frac{6}{205} \cdot 0 & \frac{6}{205} \cdot 0 \\ 0 & 0 & \frac{395}{6} & 5 & 0 \end{array}]$

Bước 1.7.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{395}{6} & 5 & 0 \end{array}]$

Bước 1.8

Perform the row operation $R_{4} = R_{4} - \frac{395}{6} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Bước 1.8.1

Perform the row operation $R_{4} = R_{4} - \frac{395}{6} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 - \frac{395}{6} \cdot 0 & 0 - \frac{395}{6} \cdot 0 & \frac{395}{6} - \frac{395}{6} \cdot 1 & 5 - \frac{395}{6} \cdot 0 & 0 - \frac{395}{6} \cdot 0 \end{array}]$

Bước 1.8.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 5 & 0 \end{array}]$

Bước 1.9

Multiply each element of $R_{4}$ by $\frac{1}{5}$ to make the entry at $4, 4$ a $1$ .

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Bước 1.9.1

Multiply each element of $R_{4}$ by $\frac{1}{5}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{5}{5} & \frac{0}{5} \end{array}]$

Bước 1.9.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & \frac{21}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.10

Perform the row operation $R_{2} = R_{2} - \frac{21}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 1.10.1

Perform the row operation $R_{2} = R_{2} - \frac{21}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 - \frac{21}{2} \cdot 0 & 1 - \frac{21}{2} \cdot 0 & \frac{21}{2} - \frac{21}{2} \cdot 1 & 0 - \frac{21}{2} \cdot 0 & 0 - \frac{21}{2} \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.10.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.11

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

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Bước 1.11.1

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

$[\begin{array}{c} 1 - \frac{1}{6} \cdot 0 & 0 - \frac{1}{6} \cdot 0 & \frac{1}{6} - \frac{1}{6} \cdot 1 & 0 - \frac{1}{6} \cdot 0 & 0 - \frac{1}{6} \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Bước 1.11.2

Rút gọn $R_{1}$ .

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

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 $4$