Lineare Algebra Beispiele

[\begin{array}{c} 6 & - 4 & a \\ 3 & - 2 & d \end{array}]

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

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{6}

$\frac{1}{6}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Schritt 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{6}

$\frac{1}{6}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{array}{c} \frac{6}{6} & \frac{- 4}{6} & \frac{a}{6} \\ 3 & - 2 & d \end{array}]

$[\begin{array}{c} \frac{6}{6} & \frac{- 4}{6} & \frac{a}{6} \\ 3 & - 2 & d \end{array}]$

Schritt 1.1.2

Vereinfache

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 3 & - 2 & d \end{array}]

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 3 & - 2 & d \end{array}]$

[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 3 & - 2 & d \end{array}]

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 3 & - 2 & d \end{array}]$

Schritt 1.2

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Schritt 1.2.1

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 3 - 3 \cdot 1 & - 2 - 3 (- \frac{2}{3}) & d - 3 \frac{a}{6} \end{array}]

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 3 - 3 \cdot 1 & - 2 - 3 (- \frac{2}{3}) & d - 3 \frac{a}{6} \end{array}]$

Schritt 1.2.2

Vereinfache

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 0 & 0 & d - \frac{a}{2} \end{array}]

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

[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ 0 & 0 & d - \frac{a}{2} \end{array}]

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

Schritt 1.3

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{d - \frac{a}{2}}

$\frac{1}{d - \frac{a}{2}}$ to make the entry at

2, 3

$2, 3$ a

1

$1$ .

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Schritt 1.3.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{d - \frac{a}{2}}

$\frac{1}{d - \frac{a}{2}}$ to make the entry at

2, 3

$2, 3$ a

1

$1$ .

[\begin{array}{c} 1 & - \frac{2}{3} & \frac{a}{6} \\ \frac{0}{d - \frac{a}{2}} & \frac{0}{d - \frac{a}{2}} & \frac{d - \frac{a}{2}}{d - \frac{a}{2}} \end{array}]

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

Schritt 1.3.2

Vereinfache

R_{2}

$R_{2}$ .

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

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

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

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

Schritt 1.4

Perform the row operation

R_{1} = R_{1} - \frac{a}{6} R_{2}

$R_{1} = R_{1} - \frac{a}{6} R_{2}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Schritt 1.4.1

Perform the row operation

R_{1} = R_{1} - \frac{a}{6} R_{2}

$R_{1} = R_{1} - \frac{a}{6} R_{2}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

[\begin{array}{c} 1 - \frac{a}{6} \cdot 0 & - \frac{2}{3} - \frac{a}{6} \cdot 0 & \frac{a}{6} - \frac{a}{6} \cdot 1 \\ 0 & 0 & 1 \end{array}]

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

Schritt 1.4.2

Vereinfache

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Schritt 2

The pivot positions are the locations with the leading

1

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

a_{11}

$a_{11}$ and

a_{23}

$a_{23}$

Pivot Columns:

1

$1$ and

3

$3$