Aljabar Linear Contoh

$[\begin{array}{c} 3 & 0 & 2 & 2 \\ - 6 & 42 & 24 & 54 \\ 21 & - 21 & 0 & - 15 \end{array}]$

Langkah 1

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 1.1

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

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Langkah 1.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{0}{3} & \frac{2}{3} & \frac{2}{3} \\ - 6 & 42 & 24 & 54 \\ 21 & - 21 & 0 & - 15 \end{array}]$

Langkah 1.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ - 6 & 42 & 24 & 54 \\ 21 & - 21 & 0 & - 15 \end{array}]$

Langkah 1.2

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

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Langkah 1.2.1

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

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ - 6 + 6 \cdot 1 & 42 + 6 \cdot 0 & 24 + 6 (\frac{2}{3}) & 54 + 6 (\frac{2}{3}) \\ 21 & - 21 & 0 & - 15 \end{array}]$

Langkah 1.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ 0 & 42 & 28 & 58 \\ 21 & - 21 & 0 & - 15 \end{array}]$

Langkah 1.3

Perform the row operation $R_{3} = R_{3} - 21 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 1.3.1

Perform the row operation $R_{3} = R_{3} - 21 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ 0 & 42 & 28 & 58 \\ 21 - 21 \cdot 1 & - 21 - 21 \cdot 0 & 0 - 21 (\frac{2}{3}) & - 15 - 21 (\frac{2}{3}) \end{array}]$

Langkah 1.3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ 0 & 42 & 28 & 58 \\ 0 & - 21 & - 14 & - 29 \end{array}]$

Langkah 1.4

Multiply each element of $R_{2}$ by $\frac{1}{42}$ to make the entry at $2, 2$ a $1$ .

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Langkah 1.4.1

Multiply each element of $R_{2}$ by $\frac{1}{42}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ \frac{0}{42} & \frac{42}{42} & \frac{28}{42} & \frac{58}{42} \\ 0 & - 21 & - 14 & - 29 \end{array}]$

Langkah 1.4.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ 0 & 1 & \frac{2}{3} & \frac{29}{21} \\ 0 & - 21 & - 14 & - 29 \end{array}]$

Langkah 1.5

Perform the row operation $R_{3} = R_{3} + 21 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Langkah 1.5.1

Perform the row operation $R_{3} = R_{3} + 21 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ 0 & 1 & \frac{2}{3} & \frac{29}{21} \\ 0 + 21 \cdot 0 & - 21 + 21 \cdot 1 & - 14 + 21 (\frac{2}{3}) & - 29 + 21 (\frac{29}{21}) \end{array}]$

Langkah 1.5.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{3} & \frac{2}{3} \\ 0 & 1 & \frac{2}{3} & \frac{29}{21} \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 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 $2$

Langkah 3

The rank is the number of pivot columns.

$2$