الجبر الخطي الأمثلة

$[\begin{array}{c} 7 & 8 \\ 5 & - 3 \\ 8 & 5 \end{array}]$

خطوة 1

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 1.1

Multiply each element of $R_{1}$ by $\frac{1}{7}$ to make the entry at $1, 1$ a $1$ .

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خطوة 1.1.1

Multiply each element of $R_{1}$ by $\frac{1}{7}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{7}{7} & \frac{8}{7} \\ 5 & - 3 \\ 8 & 5 \end{array}]$

خطوة 1.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 5 & - 3 \\ 8 & 5 \end{array}]$

خطوة 1.2

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 1.2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 5 - 5 \cdot 1 & - 3 - 5 (\frac{8}{7}) \\ 8 & 5 \end{array}]$

خطوة 1.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 0 & - \frac{61}{7} \\ 8 & 5 \end{array}]$

خطوة 1.3

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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خطوة 1.3.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 0 & - \frac{61}{7} \\ 8 - 8 \cdot 1 & 5 - 8 (\frac{8}{7}) \end{array}]$

خطوة 1.3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 0 & - \frac{61}{7} \\ 0 & - \frac{29}{7} \end{array}]$

خطوة 1.4

Multiply each element of $R_{2}$ by $- \frac{7}{61}$ to make the entry at $2, 2$ a $1$ .

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خطوة 1.4.1

Multiply each element of $R_{2}$ by $- \frac{7}{61}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ - \frac{7}{61} \cdot 0 & - \frac{7}{61} (- \frac{61}{7}) \\ 0 & - \frac{29}{7} \end{array}]$

خطوة 1.4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 0 & 1 \\ 0 & - \frac{29}{7} \end{array}]$

خطوة 1.5

Perform the row operation $R_{3} = R_{3} + \frac{29}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

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خطوة 1.5.1

Perform the row operation $R_{3} = R_{3} + \frac{29}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 0 & 1 \\ 0 + \frac{29}{7} \cdot 0 & - \frac{29}{7} + \frac{29}{7} \cdot 1 \end{array}]$

خطوة 1.5.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{8}{7} \\ 0 & 1 \\ 0 & 0 \end{array}]$

خطوة 1.6

Perform the row operation $R_{1} = R_{1} - \frac{8}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

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خطوة 1.6.1

Perform the row operation $R_{1} = R_{1} - \frac{8}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{8}{7} \cdot 0 & \frac{8}{7} - \frac{8}{7} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

خطوة 1.6.2

بسّط $R_{1}$ .

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

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: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$