Đại số Ví dụ

$[\begin{array}{c} - 2 & 0 & 4 & 0 \\ 167 & 0 & - 119 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Bước 1

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

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Bước 1.1

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

$[\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 \\ 167 & 0 & - 119 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Bước 1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 167 & 0 & - 119 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Bước 2

Perform the row operation $R_{2} = R_{2} - 167 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 2.1

Perform the row operation $R_{2} = R_{2} - 167 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 167 - 167 \cdot 1 & 0 - 167 \cdot 0 & - 119 - 167 \cdot - 2 & - 14 - 167 \cdot 0 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Bước 2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Bước 3

Perform the row operation $R_{3} = R_{3} - 15 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 3.1

Perform the row operation $R_{3} = R_{3} - 15 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 15 - 15 \cdot 1 & 70 - 15 \cdot 0 & - 15 - 15 \cdot - 2 & 0 - 15 \cdot 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Bước 3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 70 & 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Bước 4

Perform the row operation $R_{4} = R_{4} + 325 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Bước 4.1

Perform the row operation $R_{4} = R_{4} + 325 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 70 & 15 & 0 \\ - 325 + 325 \cdot 1 & 0 + 325 \cdot 0 & 225 + 325 \cdot - 2 & 100 + 325 \cdot 0 \end{array}]$

Bước 4.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 70 & 15 & 0 \\ 0 & 0 & - 425 & 100 \end{array}]$

Bước 5

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 70 & 15 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 0 & - 425 & 100 \end{array}]$

Bước 6

Multiply each element of $R_{2}$ by $\frac{1}{70}$ to make the entry at $2, 2$ a $1$ .

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Bước 6.1

Multiply each element of $R_{2}$ by $\frac{1}{70}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ \frac{0}{70} & \frac{70}{70} & \frac{15}{70} & \frac{0}{70} \\ 0 & 0 & 215 & - 14 \\ 0 & 0 & - 425 & 100 \end{array}]$

Bước 6.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 0 & - 425 & 100 \end{array}]$

Bước 7

Multiply each element of $R_{3}$ by $\frac{1}{215}$ to make the entry at $3, 3$ a $1$ .

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Bước 7.1

Multiply each element of $R_{3}$ by $\frac{1}{215}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ \frac{0}{215} & \frac{0}{215} & \frac{215}{215} & \frac{- 14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

Bước 7.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

Bước 8

Perform the row operation $R_{4} = R_{4} + 425 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Bước 8.1

Perform the row operation $R_{4} = R_{4} + 425 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 + 425 \cdot 0 & 0 + 425 \cdot 0 & - 425 + 425 \cdot 1 & 100 + 425 (- \frac{14}{215}) \end{array}]$

Bước 8.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & \frac{3110}{43} \end{array}]$

Bước 9

Multiply each element of $R_{4}$ by $\frac{43}{3110}$ to make the entry at $4, 4$ a $1$ .

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Bước 9.1

Multiply each element of $R_{4}$ by $\frac{43}{3110}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ \frac{43}{3110} \cdot 0 & \frac{43}{3110} \cdot 0 & \frac{43}{3110} \cdot 0 & \frac{43}{3110} \cdot \frac{3110}{43} \end{array}]$

Bước 9.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

Bước 10

Perform the row operation $R_{3} = R_{3} + \frac{14}{215} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Bước 10.1

Perform the row operation $R_{3} = R_{3} + \frac{14}{215} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 + \frac{14}{215} \cdot 0 & 0 + \frac{14}{215} \cdot 0 & 1 + \frac{14}{215} \cdot 0 & - \frac{14}{215} + \frac{14}{215} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Bước 10.2

Rút gọn $R_{3}$ .

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

Bước 11

Perform the row operation $R_{2} = R_{2} - \frac{3}{14} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 11.1

Perform the row operation $R_{2} = R_{2} - \frac{3}{14} R_{3}$ to make the entry at $2, 3$ a $0$ .

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

Bước 11.2

Rút gọn $R_{2}$ .

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

Bước 12

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

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Bước 12.1

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

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

Bước 12.2

Rút gọn $R_{1}$ .

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