Beispiele

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

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

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 column

2

$2$ 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_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

| \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} |

$| \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} |$

Schritt 1.4

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} |

$- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} |$

Schritt 1.5

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |

$| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |$

Schritt 1.6

Multiply element

a_{22}

$a_{22}$ by its cofactor.

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

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

Schritt 1.7

The minor for

a_{32}

$a_{32}$ is the determinant with row

3

$3$ and column

2

$2$ deleted.

| \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |

$| \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |$

Schritt 1.8

Multiply element

a_{32}

$a_{32}$ by its cofactor.

0 | \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |

$0 | \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |$

Schritt 1.9

Add the terms together.

- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} | + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |

$- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} | + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |$

- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} | + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |

$- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} | + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |$

Schritt 2

Mutltipliziere

0

$0$ mit

| \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |

$| \begin{array}{c} 1 & 1 \\ 4 & 4 \end{array} |$ .

- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} | + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0

$- 1 | \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} | + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0$

Schritt 3

Berechne

| \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} |

$| \begin{array}{c} 4 & 4 \\ 1 & 1 \end{array} |$ .

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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.

- 1 (4 \cdot 1 - 1 \cdot 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0

$- 1 (4 \cdot 1 - 1 \cdot 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0$

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

4

$4$ mit

1

$1$ .

- 1 (4 - 1 \cdot 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0

$- 1 (4 - 1 \cdot 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0$

Schritt 3.2.1.2

Mutltipliziere

- 1

$- 1$ mit

4

$4$ .

- 1 (4 - 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0

$- 1 (4 - 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0$

- 1 (4 - 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0

$- 1 (4 - 4) + 3 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0$

Schritt 3.2.2

Subtrahiere

4

$4$ von

4

$4$ .

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

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

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

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

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

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

Schritt 4

Berechne

| \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} |

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

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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.

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

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

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

1

$1$ mit

1

$1$ .

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

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

Schritt 4.2.1.2

Mutltipliziere

- 1

$- 1$ mit

1

$1$ .

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

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

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

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

Schritt 4.2.2

Subtrahiere

1

$1$ von

1

$1$ .

- 1 \cdot 0 + 3 \cdot 0 + 0

$- 1 \cdot 0 + 3 \cdot 0 + 0$

- 1 \cdot 0 + 3 \cdot 0 + 0

$- 1 \cdot 0 + 3 \cdot 0 + 0$

- 1 \cdot 0 + 3 \cdot 0 + 0

$- 1 \cdot 0 + 3 \cdot 0 + 0$

Schritt 5

Vereinfache die Determinante.

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Schritt 5.1

Vereinfache jeden Term.

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Schritt 5.1.1

Mutltipliziere

- 1

$- 1$ mit

0

$0$ .

0 + 3 \cdot 0 + 0

$0 + 3 \cdot 0 + 0$

Schritt 5.1.2

Mutltipliziere

3

$3$ mit

0

$0$ .

0 + 0 + 0

$0 + 0 + 0$

0 + 0 + 0

$0 + 0 + 0$

Schritt 5.2

Addiere

0

$0$ und

0

$0$ .

0 + 0

$0 + 0$

Schritt 5.3

Addiere

0

$0$ und

0

$0$ .

0

$0$

0

$0$

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