Lineare Algebra Beispiele

$[\begin{array}{c} 7 & 2 & 4 \\ 6 & 6 & 3 \\ 8 & 4 & 9 \end{array}]$

Schritt 1

Find the determinant.

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Schritt 1.1

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 row $1$ by its cofactor and add.

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Schritt 1.1.1

Consider the corresponding sign chart.

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

Schritt 1.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Schritt 1.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 6 & 3 \\ 4 & 9 \end{array} |$

Schritt 1.1.4

Multiply element $a_{11}$ by its cofactor.

$7 | \begin{array}{c} 6 & 3 \\ 4 & 9 \end{array} |$

Schritt 1.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} |$

Schritt 1.1.6

Multiply element $a_{12}$ by its cofactor.

$- 2 | \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} |$

Schritt 1.1.7

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

$| \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.1.8

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

$4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.1.9

Add the terms together.

$7 | \begin{array}{c} 6 & 3 \\ 4 & 9 \end{array} | - 2 | \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} | + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.2

Berechne $| \begin{array}{c} 6 & 3 \\ 4 & 9 \end{array} |$ .

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Schritt 1.2.1

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.

$7 (6 \cdot 9 - 4 \cdot 3) - 2 | \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} | + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.2.2

Vereinfache die Determinante.

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Schritt 1.2.2.1

Vereinfache jeden Term.

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Schritt 1.2.2.1.1

Mutltipliziere $6$ mit $9$ .

$7 (54 - 4 \cdot 3) - 2 | \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} | + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.2.2.1.2

Mutltipliziere $- 4$ mit $3$ .

$7 (54 - 12) - 2 | \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} | + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.2.2.2

Subtrahiere $12$ von $54$ .

$7 \cdot 42 - 2 | \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} | + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.3

Berechne $| \begin{array}{c} 6 & 3 \\ 8 & 9 \end{array} |$ .

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Schritt 1.3.1

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.

$7 \cdot 42 - 2 (6 \cdot 9 - 8 \cdot 3) + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.3.2

Vereinfache die Determinante.

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Schritt 1.3.2.1

Vereinfache jeden Term.

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Schritt 1.3.2.1.1

Mutltipliziere $6$ mit $9$ .

$7 \cdot 42 - 2 (54 - 8 \cdot 3) + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.3.2.1.2

Mutltipliziere $- 8$ mit $3$ .

$7 \cdot 42 - 2 (54 - 24) + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.3.2.2

Subtrahiere $24$ von $54$ .

$7 \cdot 42 - 2 \cdot 30 + 4 | \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$

Schritt 1.4

Berechne $| \begin{array}{c} 6 & 6 \\ 8 & 4 \end{array} |$ .

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Schritt 1.4.1

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.

$7 \cdot 42 - 2 \cdot 30 + 4 (6 \cdot 4 - 8 \cdot 6)$

Schritt 1.4.2

Vereinfache die Determinante.

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Schritt 1.4.2.1

Vereinfache jeden Term.

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Schritt 1.4.2.1.1

Mutltipliziere $6$ mit $4$ .

$7 \cdot 42 - 2 \cdot 30 + 4 (24 - 8 \cdot 6)$

Schritt 1.4.2.1.2

Mutltipliziere $- 8$ mit $6$ .

$7 \cdot 42 - 2 \cdot 30 + 4 (24 - 48)$

Schritt 1.4.2.2

Subtrahiere $48$ von $24$ .

$7 \cdot 42 - 2 \cdot 30 + 4 \cdot - 24$

Schritt 1.5

Vereinfache die Determinante.

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Schritt 1.5.1

Vereinfache jeden Term.

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Schritt 1.5.1.1

Mutltipliziere $7$ mit $42$ .

$294 - 2 \cdot 30 + 4 \cdot - 24$

Schritt 1.5.1.2

Mutltipliziere $- 2$ mit $30$ .

$294 - 60 + 4 \cdot - 24$

Schritt 1.5.1.3

Mutltipliziere $4$ mit $- 24$ .

$294 - 60 - 96$

Schritt 1.5.2

Subtrahiere $60$ von $294$ .

$234 - 96$

Schritt 1.5.3

Subtrahiere $96$ von $234$ .

$138$

Schritt 2

Since the determinant is non-zero, the inverse exists.

Schritt 3

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} 7 & 2 & 4 & 1 & 0 & 0 \\ 6 & 6 & 3 & 0 & 1 & 0 \\ 8 & 4 & 9 & 0 & 0 & 1 \end{array}]$

Schritt 4

Ermittele die normierte Zeilenstufenform.

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Schritt 4.1

Multiply each element of $R_{1}$ by $\frac{1}{7}$ to make the entry at $1, 1$ a $1$ .

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Schritt 4.1.1

Multiply each element of $R_{1}$ by $\frac{1}{7}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{7}{7} & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & \frac{0}{7} & \frac{0}{7} \\ 6 & 6 & 3 & 0 & 1 & 0 \\ 8 & 4 & 9 & 0 & 0 & 1 \end{array}]$

Schritt 4.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 6 & 6 & 3 & 0 & 1 & 0 \\ 8 & 4 & 9 & 0 & 0 & 1 \end{array}]$

Schritt 4.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 4.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 6 - 6 \cdot 1 & 6 - 6 (\frac{2}{7}) & 3 - 6 (\frac{4}{7}) & 0 - 6 (\frac{1}{7}) & 1 - 6 \cdot 0 & 0 - 6 \cdot 0 \\ 8 & 4 & 9 & 0 & 0 & 1 \end{array}]$

Schritt 4.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & \frac{30}{7} & - \frac{3}{7} & - \frac{6}{7} & 1 & 0 \\ 8 & 4 & 9 & 0 & 0 & 1 \end{array}]$

Schritt 4.3

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 4.3.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & \frac{30}{7} & - \frac{3}{7} & - \frac{6}{7} & 1 & 0 \\ 8 - 8 \cdot 1 & 4 - 8 (\frac{2}{7}) & 9 - 8 (\frac{4}{7}) & 0 - 8 (\frac{1}{7}) & 0 - 8 \cdot 0 & 1 - 8 \cdot 0 \end{array}]$

Schritt 4.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & \frac{30}{7} & - \frac{3}{7} & - \frac{6}{7} & 1 & 0 \\ 0 & \frac{12}{7} & \frac{31}{7} & - \frac{8}{7} & 0 & 1 \end{array}]$

Schritt 4.4

Multiply each element of $R_{2}$ by $\frac{7}{30}$ to make the entry at $2, 2$ a $1$ .

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Schritt 4.4.1

Multiply each element of $R_{2}$ by $\frac{7}{30}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ \frac{7}{30} \cdot 0 & \frac{7}{30} \cdot \frac{30}{7} & \frac{7}{30} (- \frac{3}{7}) & \frac{7}{30} (- \frac{6}{7}) & \frac{7}{30} \cdot 1 & \frac{7}{30} \cdot 0 \\ 0 & \frac{12}{7} & \frac{31}{7} & - \frac{8}{7} & 0 & 1 \end{array}]$

Schritt 4.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & 1 & - \frac{1}{10} & - \frac{1}{5} & \frac{7}{30} & 0 \\ 0 & \frac{12}{7} & \frac{31}{7} & - \frac{8}{7} & 0 & 1 \end{array}]$

Schritt 4.5

Perform the row operation $R_{3} = R_{3} - \frac{12}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 4.5.1

Perform the row operation $R_{3} = R_{3} - \frac{12}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & 1 & - \frac{1}{10} & - \frac{1}{5} & \frac{7}{30} & 0 \\ 0 - \frac{12}{7} \cdot 0 & \frac{12}{7} - \frac{12}{7} \cdot 1 & \frac{31}{7} - \frac{12}{7} (- \frac{1}{10}) & - \frac{8}{7} - \frac{12}{7} (- \frac{1}{5}) & 0 - \frac{12}{7} \cdot \frac{7}{30} & 1 - \frac{12}{7} \cdot 0 \end{array}]$

Schritt 4.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & 1 & - \frac{1}{10} & - \frac{1}{5} & \frac{7}{30} & 0 \\ 0 & 0 & \frac{23}{5} & - \frac{4}{5} & - \frac{2}{5} & 1 \end{array}]$

Schritt 4.6

Multiply each element of $R_{3}$ by $\frac{5}{23}$ to make the entry at $3, 3$ a $1$ .

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Schritt 4.6.1

Multiply each element of $R_{3}$ by $\frac{5}{23}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & 1 & - \frac{1}{10} & - \frac{1}{5} & \frac{7}{30} & 0 \\ \frac{5}{23} \cdot 0 & \frac{5}{23} \cdot 0 & \frac{5}{23} \cdot \frac{23}{5} & \frac{5}{23} (- \frac{4}{5}) & \frac{5}{23} (- \frac{2}{5}) & \frac{5}{23} \cdot 1 \end{array}]$

Schritt 4.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & 1 & - \frac{1}{10} & - \frac{1}{5} & \frac{7}{30} & 0 \\ 0 & 0 & 1 & - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$

Schritt 4.7

Perform the row operation $R_{2} = R_{2} + \frac{1}{10} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 4.7.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{10} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 + \frac{1}{10} \cdot 0 & 1 + \frac{1}{10} \cdot 0 & - \frac{1}{10} + \frac{1}{10} \cdot 1 & - \frac{1}{5} + \frac{1}{10} (- \frac{4}{23}) & \frac{7}{30} + \frac{1}{10} (- \frac{2}{23}) & 0 + \frac{1}{10} \cdot \frac{5}{23} \\ 0 & 0 & 1 & - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$

Schritt 4.7.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & \frac{4}{7} & \frac{1}{7} & 0 & 0 \\ 0 & 1 & 0 & - \frac{5}{23} & \frac{31}{138} & \frac{1}{46} \\ 0 & 0 & 1 & - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$

Schritt 4.8

Perform the row operation $R_{1} = R_{1} - \frac{4}{7} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 4.8.1

Perform the row operation $R_{1} = R_{1} - \frac{4}{7} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{4}{7} \cdot 0 & \frac{2}{7} - \frac{4}{7} \cdot 0 & \frac{4}{7} - \frac{4}{7} \cdot 1 & \frac{1}{7} - \frac{4}{7} (- \frac{4}{23}) & 0 - \frac{4}{7} (- \frac{2}{23}) & 0 - \frac{4}{7} \cdot \frac{5}{23} \\ 0 & 1 & 0 & - \frac{5}{23} & \frac{31}{138} & \frac{1}{46} \\ 0 & 0 & 1 & - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$

Schritt 4.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{7} & 0 & \frac{39}{161} & \frac{8}{161} & - \frac{20}{161} \\ 0 & 1 & 0 & - \frac{5}{23} & \frac{31}{138} & \frac{1}{46} \\ 0 & 0 & 1 & - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$

Schritt 4.9

Perform the row operation $R_{1} = R_{1} - \frac{2}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 4.9.1

Perform the row operation $R_{1} = R_{1} - \frac{2}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{2}{7} \cdot 0 & \frac{2}{7} - \frac{2}{7} \cdot 1 & 0 - \frac{2}{7} \cdot 0 & \frac{39}{161} - \frac{2}{7} (- \frac{5}{23}) & \frac{8}{161} - \frac{2}{7} \cdot \frac{31}{138} & - \frac{20}{161} - \frac{2}{7} \cdot \frac{1}{46} \\ 0 & 1 & 0 & - \frac{5}{23} & \frac{31}{138} & \frac{1}{46} \\ 0 & 0 & 1 & - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$

Schritt 4.9.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{7}{23} & - \frac{1}{69} & - \frac{3}{23} \\ 0 & 1 & 0 & - \frac{5}{23} & \frac{31}{138} & \frac{1}{46} \\ 0 & 0 & 1 & - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$

Schritt 5

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} \frac{7}{23} & - \frac{1}{69} & - \frac{3}{23} \\ - \frac{5}{23} & \frac{31}{138} & \frac{1}{46} \\ - \frac{4}{23} & - \frac{2}{23} & \frac{5}{23} \end{array}]$