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$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 1 & - 1 & - 7 & 7 \\ 0.75 & - 0.5 & 0.25 & 18 \\ 1 & 1 & 1 & 1 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 1 - 1 & - 1 - 1 & - 7 + 1 & 7 + 5 \\ 0.75 & - 0.5 & 0.25 & 18 \\ 1 & 1 & 1 & 1 \end{array}]$

Bước 1.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 2 & - 6 & 12 \\ 0.75 & - 0.5 & 0.25 & 18 \\ 1 & 1 & 1 & 1 \end{array}]$

Bước 2

Perform the row operation $R_{3} = R_{3} - 0.75 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 2.1

Perform the row operation $R_{3} = R_{3} - 0.75 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 2 & - 6 & 12 \\ 0.75 - 0.75 \cdot 1 & - 0.5 - 0.75 \cdot 1 & 0.25 - 0.75 \cdot - 1 & 18 - 0.75 \cdot - 5 \\ 1 & 1 & 1 & 1 \end{array}]$

Bước 2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 2 & - 6 & 12 \\ 0 & - 1.25 & 1 & 21.75 \\ 1 & 1 & 1 & 1 \end{array}]$

Bước 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 3.1

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

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 2 & - 6 & 12 \\ 0 & - 1.25 & 1 & 21.75 \\ 1 - 1 & 1 - 1 & 1 + 1 & 1 + 5 \end{array}]$

Bước 3.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 2 & - 6 & 12 \\ 0 & - 1.25 & 1 & 21.75 \\ 0 & 0 & 2 & 6 \end{array}]$

Bước 4

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

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Bước 4.1

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

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 6 & - \frac{1}{2} \cdot 12 \\ 0 & - 1.25 & 1 & 21.75 \\ 0 & 0 & 2 & 6 \end{array}]$

Bước 4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & 1 & 3 & - 6 \\ 0 & - 1.25 & 1 & 21.75 \\ 0 & 0 & 2 & 6 \end{array}]$

Bước 5

Perform the row operation $R_{3} = R_{3} + 1.25 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 5.1

Perform the row operation $R_{3} = R_{3} + 1.25 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & 1 & 3 & - 6 \\ 0 + 1.25 \cdot 0 & - 1.25 + 1.25 \cdot 1 & 1 + 1.25 \cdot 3 & 21.75 + 1.25 \cdot - 6 \\ 0 & 0 & 2 & 6 \end{array}]$

Bước 5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & 1 & 3 & - 6 \\ 0 & 0 & 4.75 & 14.25 \\ 0 & 0 & 2 & 6 \end{array}]$

Bước 6

Multiply each element of $R_{3}$ by $\frac{1}{4.75}$ to make the entry at $3, 3$ a $1$ .

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Bước 6.1

Multiply each element of $R_{3}$ by $\frac{1}{4.75}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & 1 & 3 & - 6 \\ \frac{0}{4.75} & \frac{0}{4.75} & \frac{4.75}{4.75} & \frac{14.25}{4.75} \\ 0 & 0 & 2 & 6 \end{array}]$

Bước 6.2

Rút gọn $R_{3}$ .

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

Bước 7

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

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Bước 7.1

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

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

Bước 7.2

Rút gọn $R_{4}$ .

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

Bước 8

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

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Bước 8.1

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

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

Bước 8.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & 1 & 0 & - 15 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 9

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

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Bước 9.1

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

$[\begin{array}{c} 1 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & - 5 + 1 \cdot 3 \\ 0 & 1 & 0 & - 15 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 9.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & - 2 \\ 0 & 1 & 0 & - 15 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 10

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

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Bước 10.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & - 2 + 15 \\ 0 & 1 & 0 & - 15 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 10.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 13 \\ 0 & 1 & 0 & - 15 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}]$