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$[\begin{array}{c} 1 & 1 & 2 & 3 & 1 \\ 2 & 1 & 3 & 4 & 1 \\ 3 & 1 & 4 & 5 & 2 \end{array}]$

Bước 1

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

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Bước 1.1

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

$[\begin{array}{c} 1 & 1 & 2 & 3 & 1 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 1 & 3 - 2 \cdot 2 & 4 - 2 \cdot 3 & 1 - 2 \cdot 1 \\ 3 & 1 & 4 & 5 & 2 \end{array}]$

Bước 1.2

Rút gọn $R_{2}$ .

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

Bước 2

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

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Bước 2.1

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

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

Bước 2.2

Rút gọn $R_{3}$ .

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

Bước 3

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

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Bước 3.1

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

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

Bước 3.2

Rút gọn $R_{2}$ .

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

Bước 4

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 4.1

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

$[\begin{array}{c} 1 & 1 & 2 & 3 & 1 \\ 0 & 1 & 1 & 2 & 1 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 2 + 2 \cdot 1 & - 4 + 2 \cdot 2 & - 1 + 2 \cdot 1 \end{array}]$

Bước 4.2

Rút gọn $R_{3}$ .

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

Bước 5

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 5$ a $0$ .

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Bước 5.1

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 5$ a $0$ .

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

Bước 5.2

Rút gọn $R_{2}$ .

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

Bước 6

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

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Bước 6.1

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

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

Bước 6.2

Rút gọn $R_{1}$ .

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

Bước 7

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

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Bước 7.1

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

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

Bước 7.2

Rút gọn $R_{1}$ .

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