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$[\begin{array}{c} 0 & 0 & 3 & - 9 & 6 \\ 5 & 15 & - 10 & 40 & - 45 \\ 1 & 3 & - 1 & 5 & - 7 \end{array}]$

Bước 1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 5 & 15 & - 10 & 40 & - 45 \\ 0 & 0 & 3 & - 9 & 6 \\ 1 & 3 & - 1 & 5 & - 7 \end{array}]$

Bước 2

Multiply each element of $R_{1}$ by $\frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

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Bước 2.1

Multiply each element of $R_{1}$ by $\frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{5}{5} & \frac{15}{5} & \frac{- 10}{5} & \frac{40}{5} & \frac{- 45}{5} \\ 0 & 0 & 3 & - 9 & 6 \\ 1 & 3 & - 1 & 5 & - 7 \end{array}]$

Bước 2.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 3 & - 2 & 8 & - 9 \\ 0 & 0 & 3 & - 9 & 6 \\ 1 & 3 & - 1 & 5 & - 7 \end{array}]$

Bước 3

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

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Bước 3.1

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

$[\begin{array}{c} 1 & 3 & - 2 & 8 & - 9 \\ 0 & 0 & 3 & - 9 & 6 \\ 1 - 1 & 3 - 3 & - 1 + 2 & 5 - 8 & - 7 + 9 \end{array}]$

Bước 3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 2 & 8 & - 9 \\ 0 & 0 & 3 & - 9 & 6 \\ 0 & 0 & 1 & - 3 & 2 \end{array}]$

Bước 4

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

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Bước 4.1

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

$[\begin{array}{c} 1 & 3 & - 2 & 8 & - 9 \\ \frac{0}{3} & \frac{0}{3} & \frac{3}{3} & \frac{- 9}{3} & \frac{6}{3} \\ 0 & 0 & 1 & - 3 & 2 \end{array}]$

Bước 4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - 2 & 8 & - 9 \\ 0 & 0 & 1 & - 3 & 2 \\ 0 & 0 & 1 & - 3 & 2 \end{array}]$

Bước 5

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

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Bước 5.1

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

$[\begin{array}{c} 1 & 3 & - 2 & 8 & - 9 \\ 0 & 0 & 1 & - 3 & 2 \\ 0 - 0 & 0 - 0 & 1 - 1 & - 3 + 3 & 2 - 2 \end{array}]$

Bước 5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 2 & 8 & - 9 \\ 0 & 0 & 1 & - 3 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 6

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

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Bước 6.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & 3 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 8 + 2 \cdot - 3 & - 9 + 2 \cdot 2 \\ 0 & 0 & 1 & - 3 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Bước 6.2

Rút gọn $R_{1}$ .

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