الجبر الخطي الأمثلة

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0.07 & 0.08 & 0.09 & 274 \\ 0 & - 1 & 1 & 1000 \end{array}]$

خطوة 1

Perform the row operation $R_{2} = R_{2} - 0.07 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 1.1

Perform the row operation $R_{2} = R_{2} - 0.07 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0.07 - 0.07 \cdot 1 & 0.08 - 0.07 \cdot 1 & 0.09 - 0.07 \cdot 1 & 274 - 0.07 \cdot 3200 \\ 0 & - 1 & 1 & 1000 \end{array}]$

خطوة 1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 & 0.01 & 0.02 & 50 \\ 0 & - 1 & 1 & 1000 \end{array}]$

خطوة 2

Multiply each element of $R_{2}$ by $\frac{1}{0.01}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.1

Multiply each element of $R_{2}$ by $\frac{1}{0.01}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ \frac{0}{0.01} & \frac{0.01}{0.01} & \frac{0.02}{0.01} & \frac{50}{0.01} \\ 0 & - 1 & 1 & 1000 \end{array}]$

خطوة 2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 & 1 & 2 & 5000 \\ 0 & - 1 & 1 & 1000 \end{array}]$

خطوة 3

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

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خطوة 3.1

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

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 & 1 & 2 & 5000 \\ 0 + 0 & - 1 + 1 \cdot 1 & 1 + 1 \cdot 2 & 1000 + 1 \cdot 5000 \end{array}]$

خطوة 3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 & 1 & 2 & 5000 \\ 0 & 0 & 3 & 6000 \end{array}]$

خطوة 4

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

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خطوة 4.1

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

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 & 1 & 2 & 5000 \\ \frac{0}{3} & \frac{0}{3} & \frac{3}{3} & \frac{6000}{3} \end{array}]$

خطوة 4.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 & 1 & 2 & 5000 \\ 0 & 0 & 1 & 2000 \end{array}]$

خطوة 5

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

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خطوة 5.1

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

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 5000 - 2 \cdot 2000 \\ 0 & 0 & 1 & 2000 \end{array}]$

خطوة 5.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 3200 \\ 0 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \end{array}]$

خطوة 6

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

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خطوة 6.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 3200 - 2000 \\ 0 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \end{array}]$

خطوة 6.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 1200 \\ 0 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \end{array}]$

خطوة 7

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

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خطوة 7.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 1200 - 1000 \\ 0 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \end{array}]$

خطوة 7.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 200 \\ 0 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \end{array}]$