Aljabar Linear Contoh

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

Langkah 1

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 1.1

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

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Langkah 1.1.1

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

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

Langkah 1.1.2

Sederhanakan $R_{2}$ .

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

Langkah 1.2

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Langkah 1.2.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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

Langkah 1.2.2

Sederhanakan $R_{2}$ .

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

Langkah 1.3

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

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Langkah 1.3.1

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

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

Langkah 1.3.2

Sederhanakan $R_{3}$ .

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

Langkah 1.4

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

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Langkah 1.4.1

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

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

Langkah 1.4.2

Sederhanakan $R_{3}$ .

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

Langkah 1.5

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

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Langkah 1.5.1

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

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

Langkah 1.5.2

Sederhanakan $R_{2}$ .

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

Langkah 1.6

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

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Langkah 1.6.1

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

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

Langkah 1.6.2

Sederhanakan $R_{1}$ .

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

Langkah 1.7

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

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Langkah 1.7.1

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

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

Langkah 1.7.2

Sederhanakan $R_{1}$ .

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

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: $a_{11}, a_{22},$ and $a_{33}$

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