الجبر الخطي الأمثلة

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 3 & 0 & 2 & - 2 \\ 0 & 4 & - 1 & - 1 \\ 5 & 0 & 3 & - 1 \end{array}]$

خطوة 1

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 1.1

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

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خطوة 1.1.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 3 - 3 \cdot 1 & 0 - 3 \cdot - 2 & 2 - 3 \cdot 3 & - 2 - 3 \cdot 1 \\ 0 & 4 & - 1 & - 1 \\ 5 & 0 & 3 & - 1 \end{array}]$

خطوة 1.1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 6 & - 7 & - 5 \\ 0 & 4 & - 1 & - 1 \\ 5 & 0 & 3 & - 1 \end{array}]$

خطوة 1.2

Perform the row operation $R_{4} = R_{4} - 5 R_{1}$ to make the entry at $4, 1$ a $0$ .

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خطوة 1.2.1

Perform the row operation $R_{4} = R_{4} - 5 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 6 & - 7 & - 5 \\ 0 & 4 & - 1 & - 1 \\ 5 - 5 \cdot 1 & 0 - 5 \cdot - 2 & 3 - 5 \cdot 3 & - 1 - 5 \cdot 1 \end{array}]$

خطوة 1.2.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 6 & - 7 & - 5 \\ 0 & 4 & - 1 & - 1 \\ 0 & 10 & - 12 & - 6 \end{array}]$

خطوة 1.3

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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خطوة 1.3.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ \frac{0}{6} & \frac{6}{6} & \frac{- 7}{6} & \frac{- 5}{6} \\ 0 & 4 & - 1 & - 1 \\ 0 & 10 & - 12 & - 6 \end{array}]$

خطوة 1.3.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 4 & - 1 & - 1 \\ 0 & 10 & - 12 & - 6 \end{array}]$

خطوة 1.4

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

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خطوة 1.4.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & - 1 - 4 (- \frac{7}{6}) & - 1 - 4 (- \frac{5}{6}) \\ 0 & 10 & - 12 & - 6 \end{array}]$

خطوة 1.4.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & \frac{11}{3} & \frac{7}{3} \\ 0 & 10 & - 12 & - 6 \end{array}]$

خطوة 1.5

Perform the row operation $R_{4} = R_{4} - 10 R_{2}$ to make the entry at $4, 2$ a $0$ .

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خطوة 1.5.1

Perform the row operation $R_{4} = R_{4} - 10 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & \frac{11}{3} & \frac{7}{3} \\ 0 - 10 \cdot 0 & 10 - 10 \cdot 1 & - 12 - 10 (- \frac{7}{6}) & - 6 - 10 (- \frac{5}{6}) \end{array}]$

خطوة 1.5.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & \frac{11}{3} & \frac{7}{3} \\ 0 & 0 & - \frac{1}{3} & \frac{7}{3} \end{array}]$

خطوة 1.6

Multiply each element of $R_{3}$ by $\frac{3}{11}$ to make the entry at $3, 3$ a $1$ .

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خطوة 1.6.1

Multiply each element of $R_{3}$ by $\frac{3}{11}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ \frac{3}{11} \cdot 0 & \frac{3}{11} \cdot 0 & \frac{3}{11} \cdot \frac{11}{3} & \frac{3}{11} \cdot \frac{7}{3} \\ 0 & 0 & - \frac{1}{3} & \frac{7}{3} \end{array}]$

خطوة 1.6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & 1 & \frac{7}{11} \\ 0 & 0 & - \frac{1}{3} & \frac{7}{3} \end{array}]$

خطوة 1.7

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

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خطوة 1.7.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & 1 & \frac{7}{11} \\ 0 + \frac{1}{3} \cdot 0 & 0 + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & \frac{7}{3} + \frac{1}{3} \cdot \frac{7}{11} \end{array}]$

خطوة 1.7.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & 1 & \frac{7}{11} \\ 0 & 0 & 0 & \frac{28}{11} \end{array}]$

خطوة 1.8

Multiply each element of $R_{4}$ by $\frac{11}{28}$ to make the entry at $4, 4$ a $1$ .

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خطوة 1.8.1

Multiply each element of $R_{4}$ by $\frac{11}{28}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & 1 & \frac{7}{11} \\ \frac{11}{28} \cdot 0 & \frac{11}{28} \cdot 0 & \frac{11}{28} \cdot 0 & \frac{11}{28} \cdot \frac{28}{11} \end{array}]$

خطوة 1.8.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 & 0 & 1 & \frac{7}{11} \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 1.9

Perform the row operation $R_{3} = R_{3} - \frac{7}{11} R_{4}$ to make the entry at $3, 4$ a $0$ .

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خطوة 1.9.1

Perform the row operation $R_{3} = R_{3} - \frac{7}{11} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 \\ 0 & 1 & - \frac{7}{6} & - \frac{5}{6} \\ 0 - \frac{7}{11} \cdot 0 & 0 - \frac{7}{11} \cdot 0 & 1 - \frac{7}{11} \cdot 0 & \frac{7}{11} - \frac{7}{11} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 1.9.2

بسّط $R_{3}$ .

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

خطوة 1.10

Perform the row operation $R_{2} = R_{2} + \frac{5}{6} R_{4}$ to make the entry at $2, 4$ a $0$ .

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خطوة 1.10.1

Perform the row operation $R_{2} = R_{2} + \frac{5}{6} R_{4}$ to make the entry at $2, 4$ a $0$ .

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

خطوة 1.10.2

بسّط $R_{2}$ .

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

خطوة 1.11

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

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خطوة 1.11.1

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

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

خطوة 1.11.2

بسّط $R_{1}$ .

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

خطوة 1.12

Perform the row operation $R_{2} = R_{2} + \frac{7}{6} R_{3}$ to make the entry at $2, 3$ a $0$ .

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خطوة 1.12.1

Perform the row operation $R_{2} = R_{2} + \frac{7}{6} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 0 \\ 0 + \frac{7}{6} \cdot 0 & 1 + \frac{7}{6} \cdot 0 & - \frac{7}{6} + \frac{7}{6} \cdot 1 & 0 + \frac{7}{6} \cdot 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 1.12.2

بسّط $R_{2}$ .

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

خطوة 1.13

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

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خطوة 1.13.1

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

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

خطوة 1.13.2

بسّط $R_{1}$ .

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

خطوة 1.14

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

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خطوة 1.14.1

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

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

خطوة 1.14.2

بسّط $R_{1}$ .

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

خطوة 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22}, a_{33},$ and $a_{44}$

Pivot Columns: $1, 2, 3,$ and $4$