Algebra lineare Esempi

$[\begin{array}{c} 2 & 8 \\ 4 & 1 \\ 2 & 4 \end{array}]$

Passaggio 1

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 1.1

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

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Passaggio 1.1.1

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

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

Passaggio 1.1.2

Semplifica $R_{1}$ .

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

Passaggio 1.2

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

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Passaggio 1.2.1

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

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

Passaggio 1.2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 4 \\ 0 & - 15 \\ 2 & 4 \end{array}]$

Passaggio 1.3

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

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Passaggio 1.3.1

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

$[\begin{array}{c} 1 & 4 \\ 0 & - 15 \\ 2 - 2 \cdot 1 & 4 - 2 \cdot 4 \end{array}]$

Passaggio 1.3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 4 \\ 0 & - 15 \\ 0 & - 4 \end{array}]$

Passaggio 1.4

Multiply each element of $R_{2}$ by $- \frac{1}{15}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 1.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{15}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 4 \\ - \frac{1}{15} \cdot 0 & - \frac{1}{15} \cdot - 15 \\ 0 & - 4 \end{array}]$

Passaggio 1.4.2

Semplifica $R_{2}$ .

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

Passaggio 1.5

Perform the row operation $R_{3} = R_{3} + 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Passaggio 1.5.1

Perform the row operation $R_{3} = R_{3} + 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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

Passaggio 1.5.2

Semplifica $R_{3}$ .

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

Passaggio 1.6

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

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Passaggio 1.6.1

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

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

Passaggio 1.6.2

Semplifica $R_{1}$ .

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

Passaggio 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 $2$