Lineare Algebra Beispiele

[\begin{array}{c} 2 & 5 & 0 \\ 1 & 0 & - 3 \\ 2 & - 1 & 2 \end{array}]

$[\begin{array}{c} 2 & 5 & 0 \\ 1 & 0 & - 3 \\ 2 & - 1 & 2 \end{array}]$

Schritt 1

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in row

1

$1$ by its cofactor and add.

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Schritt 1.1

Consider the corresponding sign chart.

| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Schritt 1.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

Schritt 1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

| \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} |

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

Schritt 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

2 | \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} |

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

Schritt 1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

| \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} |

$| \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} |$

Schritt 1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} |

$- 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} |$

Schritt 1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

| \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |

$| \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |$

Schritt 1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

0 | \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |

$0 | \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |$

Schritt 1.9

Add the terms together.

2 | \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} | - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |

$2 | \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} | - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |$

2 | \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} | - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |

$2 | \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} | - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |$

Schritt 2

Mutltipliziere

0

$0$ mit

| \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |

$| \begin{array}{c} 1 & 0 \\ 2 & - 1 \end{array} |$ .

2 | \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} | - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 | \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} | - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

Schritt 3

Berechne

| \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} |

$| \begin{array}{c} 0 & - 3 \\ - 1 & 2 \end{array} |$ .

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Schritt 3.1

Die Determinante einer

2 \times 2

$2 \times 2$ -Matrix kann mithilfe der Formel

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ bestimmt werden.

2 (0 \cdot 2 - - - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 (0 \cdot 2 - - - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

Schritt 3.2

Vereinfache die Determinante.

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Schritt 3.2.1

Vereinfache jeden Term.

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Schritt 3.2.1.1

Mutltipliziere

0

$0$ mit

2

$2$ .

2 (0 - - - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 (0 - - - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

Schritt 3.2.1.2

Multipliziere

- - - 3

$- - - 3$ .

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Schritt 3.2.1.2.1

Mutltipliziere

- 1

$- 1$ mit

- 3

$- 3$ .

2 (0 - 1 \cdot 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 (0 - 1 \cdot 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

Schritt 3.2.1.2.2

Mutltipliziere

- 1

$- 1$ mit

3

$3$ .

2 (0 - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 (0 - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

2 (0 - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 (0 - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

2 (0 - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 (0 - 3) - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

Schritt 3.2.2

Subtrahiere

3

$3$ von

0

$0$ .

2 \cdot - 3 - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 \cdot - 3 - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

2 \cdot - 3 - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 \cdot - 3 - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

2 \cdot - 3 - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0

$2 \cdot - 3 - 5 | \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} | + 0$

Schritt 4

Berechne

| \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} |

$| \begin{array}{c} 1 & - 3 \\ 2 & 2 \end{array} |$ .

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Schritt 4.1

Die Determinante einer

2 \times 2

$2 \times 2$ -Matrix kann mithilfe der Formel

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ bestimmt werden.

2 \cdot - 3 - 5 (1 \cdot 2 - 2 \cdot - 3) + 0

$2 \cdot - 3 - 5 (1 \cdot 2 - 2 \cdot - 3) + 0$

Schritt 4.2

Vereinfache die Determinante.

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Schritt 4.2.1

Vereinfache jeden Term.

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Schritt 4.2.1.1

Mutltipliziere

2

$2$ mit

1

$1$ .

2 \cdot - 3 - 5 (2 - 2 \cdot - 3) + 0

$2 \cdot - 3 - 5 (2 - 2 \cdot - 3) + 0$

Schritt 4.2.1.2

Mutltipliziere

- 2

$- 2$ mit

- 3

$- 3$ .

2 \cdot - 3 - 5 (2 + 6) + 0

$2 \cdot - 3 - 5 (2 + 6) + 0$

2 \cdot - 3 - 5 (2 + 6) + 0

$2 \cdot - 3 - 5 (2 + 6) + 0$

Schritt 4.2.2

Addiere

2

$2$ und

6

$6$ .

2 \cdot - 3 - 5 \cdot 8 + 0

$2 \cdot - 3 - 5 \cdot 8 + 0$

2 \cdot - 3 - 5 \cdot 8 + 0

$2 \cdot - 3 - 5 \cdot 8 + 0$

2 \cdot - 3 - 5 \cdot 8 + 0

$2 \cdot - 3 - 5 \cdot 8 + 0$

Schritt 5

Vereinfache die Determinante.

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Schritt 5.1

Vereinfache jeden Term.

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Schritt 5.1.1

Mutltipliziere

2

$2$ mit

- 3

$- 3$ .

- 6 - 5 \cdot 8 + 0

$- 6 - 5 \cdot 8 + 0$

Schritt 5.1.2

Mutltipliziere

- 5

$- 5$ mit

8

$8$ .

- 6 - 40 + 0

$- 6 - 40 + 0$

- 6 - 40 + 0

$- 6 - 40 + 0$

Schritt 5.2

Subtrahiere

40

$40$ von

- 6

$- 6$ .

- 46 + 0

$- 46 + 0$

Schritt 5.3

Addiere

- 46

$- 46$ und

0

$0$ .

- 46

$- 46$

- 46

$- 46$