Lineare Algebra Beispiele

$[\begin{array}{c} 1 & 3 & - 2 \\ 3 & 9 & - 6 \\ - 2 & - 6 & 4 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 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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Schritt 1.1.1

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

$[\begin{array}{c} 1 & 3 & - 2 \\ 3 - 3 \cdot 1 & 9 - 3 \cdot 3 & - 6 - 3 \cdot - 2 \\ - 2 & - 6 & 4 \end{array}]$

Schritt 1.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - 2 \\ 0 & 0 & 0 \\ - 2 & - 6 & 4 \end{array}]$

Schritt 1.2

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

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Schritt 1.2.1

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

$[\begin{array}{c} 1 & 3 & - 2 \\ 0 & 0 & 0 \\ - 2 + 2 \cdot 1 & - 6 + 2 \cdot 3 & 4 + 2 \cdot - 2 \end{array}]$

Schritt 1.2.2

Vereinfache $R_{3}$ .

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

Schritt 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: $a_{11}$

Pivot Columns: $1$