Lineare Algebra Beispiele

⎡ ⎢ ⎣ \begin{matrix} 4 & - 3 & 1 & 0 1 & 0 & - 2 & 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦

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

Schritt 1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{4}{4} & \frac{- 3}{4} & \frac{1}{4} & \frac{0}{4} 1 & 0 & - 2 & 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 1.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 1 & 0 & - 2 & 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 1 & 0 & - 2 & 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 2

Perform the row operation

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

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

2, 1

$2, 1$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1

Perform the row operation

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

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

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 1 - 1 & 0 + \frac{3}{4} & - 2 - \frac{1}{4} & 0 - 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 2.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & \frac{3}{4} & - \frac{9}{4} & 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ - 2 & 1 & 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & \frac{3}{4} & - \frac{9}{4} & 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ - 2 & 1 & 1 & 0 \end{array}]$

Schritt 3

Perform the row operation

R_{3} = R_{3} + 2 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.1

Perform the row operation

R_{3} = R_{3} + 2 R_{1}

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

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & \frac{3}{4} & - \frac{9}{4} & 0 - 2 + 2 \cdot 1 & 1 + 2 (- \frac{3}{4}) & 1 + 2 (\frac{1}{4}) & 0 + 2 \cdot 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ - 2 + 2 \cdot 1 & 1 + 2 (- \frac{3}{4}) & 1 + 2 (\frac{1}{4}) & 0 + 2 \cdot 0 \end{array}]$

Schritt 3.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & \frac{3}{4} & - \frac{9}{4} & 0 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & \frac{3}{4} & - \frac{9}{4} & 0 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{array}]$

Schritt 4

Multiply each element of

R_{2}

$R_{2}$ by

\frac{4}{3}

$\frac{4}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 4.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{4}{3}

$\frac{4}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \frac{4}{3} \cdot 0 & \frac{4}{3} \cdot \frac{3}{4} & \frac{4}{3} (- \frac{9}{4}) & \frac{4}{3} \cdot 0 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ \frac{4}{3} \cdot 0 & \frac{4}{3} \cdot \frac{3}{4} & \frac{4}{3} (- \frac{9}{4}) & \frac{4}{3} \cdot 0 \\ 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{array}]$

Schritt 4.2

Vereinfache

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & 1 & - 3 & 0 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & 1 & - 3 & 0 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 5

Perform the row operation

R_{3} = R_{3} + \frac{1}{2} R_{2}

$R_{3} = R_{3} + \frac{1}{2} R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1

Perform the row operation

R_{3} = R_{3} + \frac{1}{2} R_{2}

$R_{3} = R_{3} + \frac{1}{2} R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & 1 & - 3 & 0 0 + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & \frac{3}{2} + \frac{1}{2} \cdot - 3 & 0 + \frac{1}{2} \cdot 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 5.2

Vereinfache

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & 1 & - 3 & 0 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - \frac{3}{4} & \frac{1}{4} & 0 0 & 1 & - 3 & 0 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 6

Perform the row operation

R_{1} = R_{1} + \frac{3}{4} R_{2}

$R_{1} = R_{1} + \frac{3}{4} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 6.1

Perform the row operation

R_{1} = R_{1} + \frac{3}{4} R_{2}

$R_{1} = R_{1} + \frac{3}{4} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 + \frac{3}{4} \cdot 0 & - \frac{3}{4} + \frac{3}{4} \cdot 1 & \frac{1}{4} + \frac{3}{4} \cdot - 3 & 0 + \frac{3}{4} \cdot 0 0 & 1 & - 3 & 0 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Schritt 6.2

Vereinfache

R_{1}

$R_{1}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 2 & 0 0 & 1 & - 3 & 0 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & - 2 & 0 0 & 1 & - 3 & 0 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦

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