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

Bước 1

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

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Bước 1.1

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

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

Bước 1.2

Rút gọn $R_{2}$ .

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

Bước 2

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Bước 2.1

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

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

Bước 2.2

Rút gọn $R_{4}$ .

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

Bước 3

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

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

Bước 4

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

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Bước 4.1

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

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

Bước 4.2

Rút gọn $R_{4}$ .

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