Finite Mathematik Beispiele

$[\begin{array}{c} 0.5 & 0.1 & 0.2 \\ 0.3 & 0.6 & 0.4 \\ 0.2 & 0.3 & 0.4 \end{array}]$

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.

$| \begin{array}{c} 0.6 & 0.4 \\ 0.3 & 0.4 \end{array} |$

Schritt 1.1.4

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

$0.5 | \begin{array}{c} 0.6 & 0.4 \\ 0.3 & 0.4 \end{array} |$

Schritt 1.1.5

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

$| \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} |$

Schritt 1.1.6

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

$- 0.1 | \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} |$

Schritt 1.1.7

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

$| \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.1.8

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

$0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.1.9

Add the terms together.

$0.5 | \begin{array}{c} 0.6 & 0.4 \\ 0.3 & 0.4 \end{array} | - 0.1 | \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} | + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.2

Berechne $| \begin{array}{c} 0.6 & 0.4 \\ 0.3 & 0.4 \end{array} |$ .

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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.

$0.5 (0.6 \cdot 0.4 - 0.3 \cdot 0.4) - 0.1 | \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} | + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

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 $0.6$ mit $0.4$ .

$0.5 (0.24 - 0.3 \cdot 0.4) - 0.1 | \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} | + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.2.2.1.2

Mutltipliziere $- 0.3$ mit $0.4$ .

$0.5 (0.24 - 0.12) - 0.1 | \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} | + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.2.2.2

Subtrahiere $0.12$ von $0.24$ .

$0.5 \cdot 0.12 - 0.1 | \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} | + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.3

Berechne $| \begin{array}{c} 0.3 & 0.4 \\ 0.2 & 0.4 \end{array} |$ .

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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.

$0.5 \cdot 0.12 - 0.1 (0.3 \cdot 0.4 - 0.2 \cdot 0.4) + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

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 $0.3$ mit $0.4$ .

$0.5 \cdot 0.12 - 0.1 (0.12 - 0.2 \cdot 0.4) + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.3.2.1.2

Mutltipliziere $- 0.2$ mit $0.4$ .

$0.5 \cdot 0.12 - 0.1 (0.12 - 0.08) + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.3.2.2

Subtrahiere $0.08$ von $0.12$ .

$0.5 \cdot 0.12 - 0.1 \cdot 0.04 + 0.2 | \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$

Schritt 1.4

Berechne $| \begin{array}{c} 0.3 & 0.6 \\ 0.2 & 0.3 \end{array} |$ .

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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.

$0.5 \cdot 0.12 - 0.1 \cdot 0.04 + 0.2 (0.3 \cdot 0.3 - 0.2 \cdot 0.6)$

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 $0.3$ mit $0.3$ .

$0.5 \cdot 0.12 - 0.1 \cdot 0.04 + 0.2 (0.09 - 0.2 \cdot 0.6)$

Schritt 1.4.2.1.2

Mutltipliziere $- 0.2$ mit $0.6$ .

$0.5 \cdot 0.12 - 0.1 \cdot 0.04 + 0.2 (0.09 - 0.12)$

Schritt 1.4.2.2

Subtrahiere $0.12$ von $0.09$ .

$0.5 \cdot 0.12 - 0.1 \cdot 0.04 + 0.2 \cdot - 0.03$

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 $0.5$ mit $0.12$ .

$0.06 - 0.1 \cdot 0.04 + 0.2 \cdot - 0.03$

Schritt 1.5.1.2

Mutltipliziere $- 0.1$ mit $0.04$ .

$0.06 - 0.004 + 0.2 \cdot - 0.03$

Schritt 1.5.1.3

Mutltipliziere $0.2$ mit $- 0.03$ .

$0.06 - 0.004 - 0.006$

Schritt 1.5.2

Subtrahiere $0.004$ von $0.06$ .

$0.056 - 0.006$

Schritt 1.5.3

Subtrahiere $0.006$ von $0.056$ .

$0.05$

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.

$[\begin{array}{c} 0.5 & 0.1 & 0.2 & 1 & 0 & 0 \\ 0.3 & 0.6 & 0.4 & 0 & 1 & 0 \\ 0.2 & 0.3 & 0.4 & 0 & 0 & 1 \end{array}]$

Schritt 4

Ermittele die normierte Zeilenstufenform.

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Schritt 4.1

Multiply each element of $R_{1}$ by $\frac{1}{0.5}$ to make the entry at $1, 1$ a $1$ .

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Schritt 4.1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.5}{0.5} & \frac{0.1}{0.5} & \frac{0.2}{0.5} & \frac{1}{0.5} & \frac{0}{0.5} & \frac{0}{0.5} \\ 0.3 & 0.6 & 0.4 & 0 & 1 & 0 \\ 0.2 & 0.3 & 0.4 & 0 & 0 & 1 \end{array}]$

Schritt 4.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0.3 & 0.6 & 0.4 & 0 & 1 & 0 \\ 0.2 & 0.3 & 0.4 & 0 & 0 & 1 \end{array}]$

Schritt 4.2

Perform the row operation $R_{2} = R_{2} - 0.3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 4.2.1

Perform the row operation $R_{2} = R_{2} - 0.3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0.3 - 0.3 \cdot 1 & 0.6 - 0.3 \cdot 0.2 & 0.4 - 0.3 \cdot 0.4 & 0 - 0.3 \cdot 2 & 1 - 0.3 \cdot 0 & 0 - 0.3 \cdot 0 \\ 0.2 & 0.3 & 0.4 & 0 & 0 & 1 \end{array}]$

Schritt 4.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 0.54 & 0.28 & - 0.6 & 1 & 0 \\ 0.2 & 0.3 & 0.4 & 0 & 0 & 1 \end{array}]$

Schritt 4.3

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 4.3.1

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 0.54 & 0.28 & - 0.6 & 1 & 0 \\ 0.2 - 0.2 \cdot 1 & 0.3 - 0.2 \cdot 0.2 & 0.4 - 0.2 \cdot 0.4 & 0 - 0.2 \cdot 2 & 0 - 0.2 \cdot 0 & 1 - 0.2 \cdot 0 \end{array}]$

Schritt 4.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 0.54 & 0.28 & - 0.6 & 1 & 0 \\ 0 & 0.26 & 0.32 & - 0.4 & 0 & 1 \end{array}]$

Schritt 4.4

Multiply each element of $R_{2}$ by $\frac{1}{0.54}$ to make the entry at $2, 2$ a $1$ .

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Schritt 4.4.1

Multiply each element of $R_{2}$ by $\frac{1}{0.54}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ \frac{0}{0.54} & \frac{0.54}{0.54} & \frac{0.28}{0.54} & \frac{- 0.6}{0.54} & \frac{1}{0.54} & \frac{0}{0.54} \\ 0 & 0.26 & 0.32 & - 0.4 & 0 & 1 \end{array}]$

Schritt 4.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 1 & 0.\overline{518} & - 1.\overline{1} & 1.\overline{851} & 0 \\ 0 & 0.26 & 0.32 & - 0.4 & 0 & 1 \end{array}]$

Schritt 4.5

Perform the row operation $R_{3} = R_{3} - 0.26 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 4.5.1

Perform the row operation $R_{3} = R_{3} - 0.26 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 1 & 0.\overline{518} & - 1.\overline{1} & 1.\overline{851} & 0 \\ 0 - 0.26 \cdot 0 & 0.26 - 0.26 \cdot 1 & 0.32 - 0.26 \cdot 0.\overline{518} & - 0.4 - 0.26 \cdot - 1.\overline{1} & 0 - 0.26 \cdot 1.\overline{851} & 1 - 0.26 \cdot 0 \end{array}]$

Schritt 4.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 1 & 0.\overline{518} & - 1.\overline{1} & 1.\overline{851} & 0 \\ 0 & 0 & 0.\overline{185} & - 0.\overline{1} & - 0.\overline{481} & 1 \end{array}]$

Schritt 4.6

Multiply each element of $R_{3}$ by $\frac{1}{0.\overline{185}}$ to make the entry at $3, 3$ a $1$ .

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Schritt 4.6.1

Multiply each element of $R_{3}$ by $\frac{1}{0.\overline{185}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 1 & 0.\overline{518} & - 1.\overline{1} & 1.\overline{851} & 0 \\ \frac{0}{0.\overline{185}} & \frac{0}{0.\overline{185}} & \frac{0.\overline{185}}{0.\overline{185}} & \frac{- 0.\overline{1}}{0.\overline{185}} & \frac{- 0.\overline{481}}{0.\overline{185}} & \frac{1}{0.\overline{185}} \end{array}]$

Schritt 4.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 1 & 0.\overline{518} & - 1.\overline{1} & 1.\overline{851} & 0 \\ 0 & 0 & 1 & - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$

Schritt 4.7

Perform the row operation $R_{2} = R_{2} - 0.\overline{518} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 4.7.1

Perform the row operation $R_{2} = R_{2} - 0.\overline{518} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 - 0.\overline{518} \cdot 0 & 1 - 0.\overline{518} \cdot 0 & 0.\overline{518} - 0.\overline{518} \cdot 1 & - 1.\overline{1} - 0.\overline{518} \cdot - 0.5\overline{9} & 1.\overline{851} - 0.\overline{518} \cdot - 2.5\overline{9} & 0 - 0.\overline{518} \cdot 5.3\overline{9} \\ 0 & 0 & 1 & - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$

Schritt 4.7.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0.2 & 0.4 & 2 & 0 & 0 \\ 0 & 1 & 0 & - 0.8 & 3.1\overline{9} & - 2.7\overline{9} \\ 0 & 0 & 1 & - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$

Schritt 4.8

Perform the row operation $R_{1} = R_{1} - 0.4 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 4.8.1

Perform the row operation $R_{1} = R_{1} - 0.4 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0.4 \cdot 0 & 0.2 - 0.4 \cdot 0 & 0.4 - 0.4 \cdot 1 & 2 - 0.4 \cdot - 0.5\overline{9} & 0 - 0.4 \cdot - 2.5\overline{9} & 0 - 0.4 \cdot 5.3\overline{9} \\ 0 & 1 & 0 & - 0.8 & 3.1\overline{9} & - 2.7\overline{9} \\ 0 & 0 & 1 & - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$

Schritt 4.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0.2 & 0 & 2.24 & 1.04 & - 2.15\overline{9} \\ 0 & 1 & 0 & - 0.8 & 3.1\overline{9} & - 2.7\overline{9} \\ 0 & 0 & 1 & - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$

Schritt 4.9

Perform the row operation $R_{1} = R_{1} - 0.2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 4.9.1

Perform the row operation $R_{1} = R_{1} - 0.2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0.2 \cdot 0 & 0.2 - 0.2 \cdot 1 & 0 - 0.2 \cdot 0 & 2.24 - 0.2 \cdot - 0.8 & 1.04 - 0.2 \cdot 3.1\overline{9} & - 2.15\overline{9} - 0.2 \cdot - 2.7\overline{9} \\ 0 & 1 & 0 & - 0.8 & 3.1\overline{9} & - 2.7\overline{9} \\ 0 & 0 & 1 & - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$

Schritt 4.9.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 2.4 & 0.4 & - 1.5\overline{9} \\ 0 & 1 & 0 & - 0.8 & 3.1\overline{9} & - 2.7\overline{9} \\ 0 & 0 & 1 & - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$

Schritt 5

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

$[\begin{array}{c} 2.4 & 0.4 & - 1.5\overline{9} \\ - 0.8 & 3.1\overline{9} & - 2.7\overline{9} \\ - 0.5\overline{9} & - 2.5\overline{9} & 5.3\overline{9} \end{array}]$