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$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 1 & 1 & 1 & 310 \\ 35000 & 70000 & 60000 & 15000000 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 1 - 1 & 1 + 2 & 1 - 0 & 310 - 0 \\ 35000 & 70000 & 60000 & 15000000 \end{array}]$

Bước 1.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 3 & 1 & 310 \\ 35000 & 70000 & 60000 & 15000000 \end{array}]$

Bước 2

Perform the row operation $R_{3} = R_{3} - 35000 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 2.1

Perform the row operation $R_{3} = R_{3} - 35000 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 3 & 1 & 310 \\ 35000 - 35000 \cdot 1 & 70000 - 35000 \cdot - 2 & 60000 - 35000 \cdot 0 & 15000000 - 35000 \cdot 0 \end{array}]$

Bước 2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 3 & 1 & 310 \\ 0 & 140000 & 60000 & 15000000 \end{array}]$

Bước 3

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

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Bước 3.1

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

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ \frac{0}{3} & \frac{3}{3} & \frac{1}{3} & \frac{310}{3} \\ 0 & 140000 & 60000 & 15000000 \end{array}]$

Bước 3.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{310}{3} \\ 0 & 140000 & 60000 & 15000000 \end{array}]$

Bước 4

Perform the row operation $R_{3} = R_{3} - 140000 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 4.1

Perform the row operation $R_{3} = R_{3} - 140000 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{310}{3} \\ 0 - 140000 \cdot 0 & 140000 - 140000 \cdot 1 & 60000 - 140000 (\frac{1}{3}) & 15000000 - 140000 (\frac{310}{3}) \end{array}]$

Bước 4.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{310}{3} \\ 0 & 0 & \frac{40000}{3} & \frac{1600000}{3} \end{array}]$

Bước 5

Multiply each element of $R_{3}$ by $\frac{3}{40000}$ to make the entry at $3, 3$ a $1$ .

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Bước 5.1

Multiply each element of $R_{3}$ by $\frac{3}{40000}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{310}{3} \\ \frac{3}{40000} \cdot 0 & \frac{3}{40000} \cdot 0 & \frac{3}{40000} \cdot \frac{40000}{3} & \frac{3}{40000} \cdot \frac{1600000}{3} \end{array}]$

Bước 5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{310}{3} \\ 0 & 0 & 1 & 40 \end{array}]$

Bước 6

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

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Bước 6.1

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

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 - \frac{1}{3} \cdot 0 & 1 - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & \frac{310}{3} - \frac{1}{3} \cdot 40 \\ 0 & 0 & 1 & 40 \end{array}]$

Bước 6.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & 0 & 90 \\ 0 & 0 & 1 & 40 \end{array}]$

Bước 7

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

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Bước 7.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & 0 + 2 \cdot 90 \\ 0 & 1 & 0 & 90 \\ 0 & 0 & 1 & 40 \end{array}]$

Bước 7.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 180 \\ 0 & 1 & 0 & 90 \\ 0 & 0 & 1 & 40 \end{array}]$