Toán hữu hạn Ví dụ

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ - 1 & 0.2 & 0.9 & 0 \\ 0.5 & - 1 & 0.1 & 0 \\ 0.5 & 0.8 & - 1 & 0 \end{array}]$

Bước 1

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

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Bước 1.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ - 1 + 1 \cdot 1 & 0.2 + 1 \cdot 1 & 0.9 + 1 \cdot 1 & 0 + 1 \cdot 1 \\ 0.5 & - 1 & 0.1 & 0 \\ 0.5 & 0.8 & - 1 & 0 \end{array}]$

Bước 1.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1.2 & 1.9 & 1 \\ 0.5 & - 1 & 0.1 & 0 \\ 0.5 & 0.8 & - 1 & 0 \end{array}]$

Bước 2

Perform the row operation $R_{3} = R_{3} - 0.5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 2.1

Perform the row operation $R_{3} = R_{3} - 0.5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1.2 & 1.9 & 1 \\ 0.5 - 0.5 \cdot 1 & - 1 - 0.5 \cdot 1 & 0.1 - 0.5 \cdot 1 & 0 - 0.5 \cdot 1 \\ 0.5 & 0.8 & - 1 & 0 \end{array}]$

Bước 2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1.2 & 1.9 & 1 \\ 0 & - 1.5 & - 0.4 & - 0.5 \\ 0.5 & 0.8 & - 1 & 0 \end{array}]$

Bước 3

Perform the row operation $R_{4} = R_{4} - 0.5 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Bước 3.1

Perform the row operation $R_{4} = R_{4} - 0.5 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1.2 & 1.9 & 1 \\ 0 & - 1.5 & - 0.4 & - 0.5 \\ 0.5 - 0.5 \cdot 1 & 0.8 - 0.5 \cdot 1 & - 1 - 0.5 \cdot 1 & 0 - 0.5 \cdot 1 \end{array}]$

Bước 3.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1.2 & 1.9 & 1 \\ 0 & - 1.5 & - 0.4 & - 0.5 \\ 0 & 0.3 & - 1.5 & - 0.5 \end{array}]$

Bước 4

Multiply each element of $R_{2}$ by $\frac{1}{1.2}$ to make the entry at $2, 2$ a $1$ .

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Bước 4.1

Multiply each element of $R_{2}$ by $\frac{1}{1.2}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ \frac{0}{1.2} & \frac{1.2}{1.2} & \frac{1.9}{1.2} & \frac{1}{1.2} \\ 0 & - 1.5 & - 0.4 & - 0.5 \\ 0 & 0.3 & - 1.5 & - 0.5 \end{array}]$

Bước 4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 & - 1.5 & - 0.4 & - 0.5 \\ 0 & 0.3 & - 1.5 & - 0.5 \end{array}]$

Bước 5

Perform the row operation $R_{3} = R_{3} + 1.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 5.1

Perform the row operation $R_{3} = R_{3} + 1.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 + 1.5 \cdot 0 & - 1.5 + 1.5 \cdot 1 & - 0.4 + 1.5 \cdot 1.58\overline{3} & - 0.5 + 1.5 \cdot 0.8\overline{3} \\ 0 & 0.3 & - 1.5 & - 0.5 \end{array}]$

Bước 5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 & 0 & 1.975 & 0.75 \\ 0 & 0.3 & - 1.5 & - 0.5 \end{array}]$

Bước 6

Perform the row operation $R_{4} = R_{4} - 0.3 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Bước 6.1

Perform the row operation $R_{4} = R_{4} - 0.3 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 & 0 & 1.975 & 0.75 \\ 0 - 0.3 \cdot 0 & 0.3 - 0.3 \cdot 1 & - 1.5 - 0.3 \cdot 1.58\overline{3} & - 0.5 - 0.3 \cdot 0.8\overline{3} \end{array}]$

Bước 6.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 & 0 & 1.975 & 0.75 \\ 0 & 0 & - 1.975 & - 0.75 \end{array}]$

Bước 7

Multiply each element of $R_{3}$ by $\frac{1}{1.975}$ to make the entry at $3, 3$ a $1$ .

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Bước 7.1

Multiply each element of $R_{3}$ by $\frac{1}{1.975}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ \frac{0}{1.975} & \frac{0}{1.975} & \frac{1.975}{1.975} & \frac{0.75}{1.975} \\ 0 & 0 & - 1.975 & - 0.75 \end{array}]$

Bước 7.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & - 1.975 & - 0.75 \end{array}]$

Bước 8

Perform the row operation $R_{4} = R_{4} + 1.975 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Bước 8.1

Perform the row operation $R_{4} = R_{4} + 1.975 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 & 0 & 1 & 0.37974683 \\ 0 + 1.975 \cdot 0 & 0 + 1.975 \cdot 0 & - 1.975 + 1.975 \cdot 1 & - 0.75 + 1.975 \cdot 0.37974683 \end{array}]$

Bước 8.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 1.58\overline{3} & 0.8\overline{3} \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 9

Perform the row operation $R_{2} = R_{2} - 1.58\overline{3} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 9.1

Perform the row operation $R_{2} = R_{2} - 1.58\overline{3} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 - 1.58\overline{3} \cdot 0 & 1 - 1.58\overline{3} \cdot 0 & 1.58\overline{3} - 1.58\overline{3} \cdot 1 & 0.8\overline{3} - 1.58\overline{3} \cdot 0.37974683 \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 9.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0.23206751 \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 10

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

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Bước 10.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 1 - 0.37974683 \\ 0 & 1 & 0 & 0.23206751 \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 10.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 0.62025316 \\ 0 & 1 & 0 & 0.23206751 \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 11

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

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Bước 11.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 + 0 & 0.62025316 - 0.23206751 \\ 0 & 1 & 0 & 0.23206751 \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 11.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0.38818565 \\ 0 & 1 & 0 & 0.23206751 \\ 0 & 0 & 1 & 0.37974683 \\ 0 & 0 & 0 & 0 \end{array}]$