Algebra lineare Esempi

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

Passaggio 1

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 1.1

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

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Passaggio 1.1.1

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

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

Passaggio 1.1.2

Semplifica $R_{2}$ .

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

Passaggio 1.2

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

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Passaggio 1.2.1

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

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

Passaggio 1.2.2

Semplifica $R_{2}$ .

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

Passaggio 1.3

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

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Passaggio 1.3.1

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

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

Passaggio 1.3.2

Semplifica $R_{1}$ .

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

Passaggio 1.4

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

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Passaggio 1.4.1

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

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

Passaggio 1.4.2

Semplifica $R_{1}$ .

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

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

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

Passaggio 3

The rank is the number of pivot columns.

$3$