Beispiele

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

Schritt 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $3$ by its cofactor and add.

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Schritt 1.1

Consider the corresponding sign chart.

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

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 $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

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

Schritt 1.4

Multiply element $a_{13}$ by its cofactor.

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

Schritt 1.5

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

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

Schritt 1.6

Multiply element $a_{23}$ by its cofactor.

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

Schritt 1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

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

Schritt 1.8

Multiply element $a_{33}$ by its cofactor.

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

Schritt 1.9

Add the terms together.

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

Schritt 2

Mutltipliziere $0$ mit $| \begin{array}{c} 0 & 1 \\ 1 & 1 \end{array} |$ .

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

Schritt 3

Mutltipliziere $0$ mit $| \begin{array}{c} 0 & 1 \\ 1 & 1 \end{array} |$ .

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

Schritt 4

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

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Schritt 4.1

Die Determinante einer $2 \times 2$ -Matrix kann mithilfe der Formel $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ bestimmt werden.

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

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 $1$ mit $1$ .

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

Schritt 4.2.1.2

Mutltipliziere $- 1$ mit $1$ .

$2 (1 - 1) + 0 + 0$

Schritt 4.2.2

Subtrahiere $1$ von $1$ .

$2 \cdot 0 + 0 + 0$

Schritt 5

Vereinfache die Determinante.

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Schritt 5.1

Mutltipliziere $2$ mit $0$ .

$0 + 0 + 0$

Schritt 5.2

Addiere $0$ und $0$ .

$0 + 0$

Schritt 5.3

Addiere $0$ und $0$ .

$0$

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