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$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 5 \\ 3 & 1 & 11 & 22 \end{array}]$

خطوة 1

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

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خطوة 1.1

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

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 5 \\ 3 - 3 \cdot 1 & 1 - 3 \cdot 1 & 11 - 3 \cdot 5 & 22 - 3 \cdot 11 \end{array}]$

خطوة 1.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 5 \\ 0 & - 2 & - 4 & - 11 \end{array}]$

خطوة 2

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 2.1

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

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 5 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 4 + 2 \cdot 2 & - 11 + 2 \cdot 5 \end{array}]$

خطوة 2.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 0 & - 1 \end{array}]$

خطوة 3

Multiply each element of $R_{3}$ by $- 1$ to make the entry at $3, 4$ a $1$ .

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خطوة 3.1

Multiply each element of $R_{3}$ by $- 1$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 5 \\ - 0 & - 0 & - 0 & - - 1 \end{array}]$

خطوة 3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 4

Perform the row operation $R_{2} = R_{2} - 5 R_{3}$ to make the entry at $2, 4$ a $0$ .

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خطوة 4.1

Perform the row operation $R_{2} = R_{2} - 5 R_{3}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 - 5 \cdot 0 & 1 - 5 \cdot 0 & 2 - 5 \cdot 0 & 5 - 5 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 5 & 11 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 5

Perform the row operation $R_{1} = R_{1} - 11 R_{3}$ to make the entry at $1, 4$ a $0$ .

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خطوة 5.1

Perform the row operation $R_{1} = R_{1} - 11 R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 11 \cdot 0 & 1 - 11 \cdot 0 & 5 - 11 \cdot 0 & 11 - 11 \cdot 1 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 5.2

بسّط $R_{1}$ .

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

خطوة 6

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

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خطوة 6.1

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

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

خطوة 6.2

بسّط $R_{1}$ .

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