Aljabar Linear Contoh

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

Langkah 1

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 1.1

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

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Langkah 1.1.1

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

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

Langkah 1.1.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 9 & - 4 \\ 5 & - 2 & 2 \end{array}]$

Langkah 1.2

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 1.2.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 9 & - 4 \\ 5 - 5 \cdot 1 & - 2 - 5 \cdot - 4 & 2 - 5 \cdot 2 \end{array}]$

Langkah 1.2.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 9 & - 4 \\ 0 & 18 & - 8 \end{array}]$

Langkah 1.3

Multiply each element of $R_{2}$ by $\frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

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Langkah 1.3.1

Multiply each element of $R_{2}$ by $\frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ \frac{0}{9} & \frac{9}{9} & \frac{- 4}{9} \\ 0 & 18 & - 8 \end{array}]$

Langkah 1.3.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & - \frac{4}{9} \\ 0 & 18 & - 8 \end{array}]$

Langkah 1.4

Perform the row operation $R_{3} = R_{3} - 18 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Langkah 1.4.1

Perform the row operation $R_{3} = R_{3} - 18 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & - \frac{4}{9} \\ 0 - 18 \cdot 0 & 18 - 18 \cdot 1 & - 8 - 18 (- \frac{4}{9}) \end{array}]$

Langkah 1.4.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 1 & - \frac{4}{9} \\ 0 & 0 & 0 \end{array}]$

Langkah 1.5

Perform the row operation $R_{1} = R_{1} + 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Langkah 1.5.1

Perform the row operation $R_{1} = R_{1} + 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 4 \cdot 0 & - 4 + 4 \cdot 1 & 2 + 4 (- \frac{4}{9}) \\ 0 & 1 & - \frac{4}{9} \\ 0 & 0 & 0 \end{array}]$

Langkah 1.5.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{2}{9} \\ 0 & 1 & - \frac{4}{9} \\ 0 & 0 & 0 \end{array}]$

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.

$2$