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$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 1 & - 1 & 2 & 7 \\ 2 & 0 & 3 & 14 \end{array}]$

خطوة 1

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

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خطوة 1.1

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

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 1 - 1 & - 1 - 1 & 2 - 1 & 7 - 7 \\ 2 & 0 & 3 & 14 \end{array}]$

خطوة 1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 0 & - 2 & 1 & 0 \\ 2 & 0 & 3 & 14 \end{array}]$

خطوة 2

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

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خطوة 2.1

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

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 0 & - 2 & 1 & 0 \\ 2 - 2 \cdot 1 & 0 - 2 \cdot 1 & 3 - 2 \cdot 1 & 14 - 2 \cdot 7 \end{array}]$

خطوة 2.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 0 & - 2 & 1 & 0 \\ 0 & - 2 & 1 & 0 \end{array}]$

خطوة 3

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

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خطوة 3.1

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

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 1 & - \frac{1}{2} \cdot 0 \\ 0 & - 2 & 1 & 0 \end{array}]$

خطوة 3.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 0 & 1 & - \frac{1}{2} & 0 \\ 0 & - 2 & 1 & 0 \end{array}]$

خطوة 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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خطوة 4.1

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

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 0 & 1 & - \frac{1}{2} & 0 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 1 + 2 (- \frac{1}{2}) & 0 + 2 \cdot 0 \end{array}]$

خطوة 4.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 7 \\ 0 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

خطوة 5

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

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خطوة 5.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 1 + \frac{1}{2} & 7 - 0 \\ 0 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

خطوة 5.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{3}{2} & 7 \\ 0 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$