الرياضيات المتناهية الأمثلة

$[\begin{array}{c} - 25 & - 6 & - 1 & - 71 \\ - 15 & - 10 & - 2 & - 12 \\ 20 & 6 & 1 & 51 \end{array}]$

خطوة 1

Multiply each element of $R_{1}$ by $- \frac{1}{25}$ to make the entry at $1, 1$ a $1$ .

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خطوة 1.1

Multiply each element of $R_{1}$ by $- \frac{1}{25}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{25} \cdot - 25 & - \frac{1}{25} \cdot - 6 & - \frac{1}{25} \cdot - 1 & - \frac{1}{25} \cdot - 71 \\ - 15 & - 10 & - 2 & - 12 \\ 20 & 6 & 1 & 51 \end{array}]$

خطوة 1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ - 15 & - 10 & - 2 & - 12 \\ 20 & 6 & 1 & 51 \end{array}]$

خطوة 2

Perform the row operation $R_{2} = R_{2} + 15 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 2.1

Perform the row operation $R_{2} = R_{2} + 15 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ - 15 + 15 \cdot 1 & - 10 + 15 (\frac{6}{25}) & - 2 + 15 (\frac{1}{25}) & - 12 + 15 (\frac{71}{25}) \\ 20 & 6 & 1 & 51 \end{array}]$

خطوة 2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & - \frac{32}{5} & - \frac{7}{5} & \frac{153}{5} \\ 20 & 6 & 1 & 51 \end{array}]$

خطوة 3

Perform the row operation $R_{3} = R_{3} - 20 R_{1}$ to make the entry at $3, 1$ a $0$ .

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خطوة 3.1

Perform the row operation $R_{3} = R_{3} - 20 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & - \frac{32}{5} & - \frac{7}{5} & \frac{153}{5} \\ 20 - 20 \cdot 1 & 6 - 20 (\frac{6}{25}) & 1 - 20 (\frac{1}{25}) & 51 - 20 (\frac{71}{25}) \end{array}]$

خطوة 3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & - \frac{32}{5} & - \frac{7}{5} & \frac{153}{5} \\ 0 & \frac{6}{5} & \frac{1}{5} & - \frac{29}{5} \end{array}]$

خطوة 4

Multiply each element of $R_{2}$ by $- \frac{5}{32}$ to make the entry at $2, 2$ a $1$ .

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خطوة 4.1

Multiply each element of $R_{2}$ by $- \frac{5}{32}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ - \frac{5}{32} \cdot 0 & - \frac{5}{32} (- \frac{32}{5}) & - \frac{5}{32} (- \frac{7}{5}) & - \frac{5}{32} \cdot \frac{153}{5} \\ 0 & \frac{6}{5} & \frac{1}{5} & - \frac{29}{5} \end{array}]$

خطوة 4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & 1 & \frac{7}{32} & - \frac{153}{32} \\ 0 & \frac{6}{5} & \frac{1}{5} & - \frac{29}{5} \end{array}]$

خطوة 5

Perform the row operation $R_{3} = R_{3} - \frac{6}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

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خطوة 5.1

Perform the row operation $R_{3} = R_{3} - \frac{6}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & 1 & \frac{7}{32} & - \frac{153}{32} \\ 0 - \frac{6}{5} \cdot 0 & \frac{6}{5} - \frac{6}{5} \cdot 1 & \frac{1}{5} - \frac{6}{5} \cdot \frac{7}{32} & - \frac{29}{5} - \frac{6}{5} (- \frac{153}{32}) \end{array}]$

خطوة 5.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & 1 & \frac{7}{32} & - \frac{153}{32} \\ 0 & 0 & - \frac{1}{16} & - \frac{1}{16} \end{array}]$

خطوة 6

Multiply each element of $R_{3}$ by $- 16$ to make the entry at $3, 3$ a $1$ .

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خطوة 6.1

Multiply each element of $R_{3}$ by $- 16$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & 1 & \frac{7}{32} & - \frac{153}{32} \\ - 16 \cdot 0 & - 16 \cdot 0 & - 16 (- \frac{1}{16}) & - 16 (- \frac{1}{16}) \end{array}]$

خطوة 6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & 1 & \frac{7}{32} & - \frac{153}{32} \\ 0 & 0 & 1 & 1 \end{array}]$

خطوة 7

Perform the row operation $R_{2} = R_{2} - \frac{7}{32} R_{3}$ to make the entry at $2, 3$ a $0$ .

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خطوة 7.1

Perform the row operation $R_{2} = R_{2} - \frac{7}{32} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 - \frac{7}{32} \cdot 0 & 1 - \frac{7}{32} \cdot 0 & \frac{7}{32} - \frac{7}{32} \cdot 1 & - \frac{153}{32} - \frac{7}{32} \cdot 1 \\ 0 & 0 & 1 & 1 \end{array}]$

خطوة 7.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & \frac{1}{25} & \frac{71}{25} \\ 0 & 1 & 0 & - 5 \\ 0 & 0 & 1 & 1 \end{array}]$

خطوة 8

Perform the row operation $R_{1} = R_{1} - \frac{1}{25} R_{3}$ to make the entry at $1, 3$ a $0$ .

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خطوة 8.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{25} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{25} \cdot 0 & \frac{6}{25} - \frac{1}{25} \cdot 0 & \frac{1}{25} - \frac{1}{25} \cdot 1 & \frac{71}{25} - \frac{1}{25} \cdot 1 \\ 0 & 1 & 0 & - 5 \\ 0 & 0 & 1 & 1 \end{array}]$

خطوة 8.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{6}{25} & 0 & \frac{14}{5} \\ 0 & 1 & 0 & - 5 \\ 0 & 0 & 1 & 1 \end{array}]$

خطوة 9

Perform the row operation $R_{1} = R_{1} - \frac{6}{25} R_{2}$ to make the entry at $1, 2$ a $0$ .

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خطوة 9.1

Perform the row operation $R_{1} = R_{1} - \frac{6}{25} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{6}{25} \cdot 0 & \frac{6}{25} - \frac{6}{25} \cdot 1 & 0 - \frac{6}{25} \cdot 0 & \frac{14}{5} - \frac{6}{25} \cdot - 5 \\ 0 & 1 & 0 & - 5 \\ 0 & 0 & 1 & 1 \end{array}]$

خطوة 9.2

بسّط $R_{1}$ .

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