الجبر الخطي الأمثلة

$[\begin{array}{c} 6 & - 4 & a \\ 3 & - 2 & d \end{array}]$

خطوة 1

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 1.1

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

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خطوة 1.1.1

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

$[\begin{array}{c} \frac{6}{6} & \frac{- 4}{6} & \frac{a}{6} \\ 3 & - 2 & d \end{array}]$

خطوة 1.1.2

بسّط $R_{1}$ .

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

خطوة 1.2

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 1.2.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 3 - 3 \cdot 1 & - 2 - 3 (- \frac{2}{3}) & d - 3 \frac{a}{6} \end{array}]$

خطوة 1.2.2

بسّط $R_{2}$ .

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

خطوة 1.3

Multiply each element of $R_{2}$ by $\frac{1}{d - \frac{a}{2}}$ to make the entry at $2, 3$ a $1$ .

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خطوة 1.3.1

Multiply each element of $R_{2}$ by $\frac{1}{d - \frac{a}{2}}$ to make the entry at $2, 3$ a $1$ .

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

خطوة 1.3.2

بسّط $R_{2}$ .

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

خطوة 1.4

Perform the row operation $R_{1} = R_{1} - \frac{a}{6} R_{2}$ to make the entry at $1, 3$ a $0$ .

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خطوة 1.4.1

Perform the row operation $R_{1} = R_{1} - \frac{a}{6} R_{2}$ to make the entry at $1, 3$ a $0$ .

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

خطوة 1.4.2

بسّط $R_{1}$ .

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

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: $a_{11}$ and $a_{23}$

Pivot Columns: $1$ and $3$