Elementarmathematik Beispiele

$[\begin{array}{c} 2 & 3 & - 1 \\ - 1 & 0 & 4 \\ 0 & 1 & 1 \end{array}]$

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 column $1$ 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.

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

Schritt 1.1.4

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

$2 | \begin{array}{c} 0 & 4 \\ 1 & 1 \end{array} |$

Schritt 1.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Schritt 1.1.6

Multiply element $a_{21}$ by its cofactor.

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

Schritt 1.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

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

Schritt 1.1.8

Multiply element $a_{31}$ by its cofactor.

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

Schritt 1.1.9

Add the terms together.

$2 | \begin{array}{c} 0 & 4 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 3 & - 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 3 & - 1 \\ 0 & 4 \end{array} |$

Schritt 1.2

Mutltipliziere $0$ mit $| \begin{array}{c} 3 & - 1 \\ 0 & 4 \end{array} |$ .

$2 | \begin{array}{c} 0 & 4 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 3 & - 1 \\ 1 & 1 \end{array} | + 0$

Schritt 1.3

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

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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.

$2 (0 \cdot 1 - 1 \cdot 4) + 1 | \begin{array}{c} 3 & - 1 \\ 1 & 1 \end{array} | + 0$

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 $0$ mit $1$ .

$2 (0 - 1 \cdot 4) + 1 | \begin{array}{c} 3 & - 1 \\ 1 & 1 \end{array} | + 0$

Schritt 1.3.2.1.2

Mutltipliziere $- 1$ mit $4$ .

$2 (0 - 4) + 1 | \begin{array}{c} 3 & - 1 \\ 1 & 1 \end{array} | + 0$

Schritt 1.3.2.2

Subtrahiere $4$ von $0$ .

$2 \cdot - 4 + 1 | \begin{array}{c} 3 & - 1 \\ 1 & 1 \end{array} | + 0$

Schritt 1.4

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

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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.

$2 \cdot - 4 + 1 (3 \cdot 1 - 1 \cdot - 1) + 0$

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 $3$ mit $1$ .

$2 \cdot - 4 + 1 (3 - 1 \cdot - 1) + 0$

Schritt 1.4.2.1.2

Mutltipliziere $- 1$ mit $- 1$ .

$2 \cdot - 4 + 1 (3 + 1) + 0$

Schritt 1.4.2.2

Addiere $3$ und $1$ .

$2 \cdot - 4 + 1 \cdot 4 + 0$

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 $2$ mit $- 4$ .

$- 8 + 1 \cdot 4 + 0$

Schritt 1.5.1.2

Mutltipliziere $4$ mit $1$ .

$- 8 + 4 + 0$

Schritt 1.5.2

Addiere $- 8$ und $4$ .

$- 4 + 0$

Schritt 1.5.3

Addiere $- 4$ und $0$ .

$- 4$

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.

$[\begin{array}{c} 2 & 3 & - 1 & 1 & 0 & 0 \\ - 1 & 0 & 4 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Schritt 4

Ermittele die normierte Zeilenstufenform.

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Schritt 4.1

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

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Schritt 4.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{3}{2} & \frac{- 1}{2} & \frac{1}{2} & \frac{0}{2} & \frac{0}{2} \\ - 1 & 0 & 4 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Schritt 4.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ - 1 & 0 & 4 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Schritt 4.2

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

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Schritt 4.2.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ - 1 + 1 \cdot 1 & 0 + \frac{3}{2} & 4 - \frac{1}{2} & 0 + \frac{1}{2} & 1 + 0 & 0 + 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Schritt 4.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & \frac{3}{2} & \frac{7}{2} & \frac{1}{2} & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Schritt 4.3

Multiply each element of $R_{2}$ by $\frac{2}{3}$ to make the entry at $2, 2$ a $1$ .

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Schritt 4.3.1

Multiply each element of $R_{2}$ by $\frac{2}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ \frac{2}{3} \cdot 0 & \frac{2}{3} \cdot \frac{3}{2} & \frac{2}{3} \cdot \frac{7}{2} & \frac{2}{3} \cdot \frac{1}{2} & \frac{2}{3} \cdot 1 & \frac{2}{3} \cdot 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Schritt 4.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & \frac{7}{3} & \frac{1}{3} & \frac{2}{3} & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{array}]$

Schritt 4.4

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

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Schritt 4.4.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & \frac{7}{3} & \frac{1}{3} & \frac{2}{3} & 0 \\ 0 - 0 & 1 - 1 & 1 - \frac{7}{3} & 0 - \frac{1}{3} & 0 - \frac{2}{3} & 1 - 0 \end{array}]$

Schritt 4.4.2

Vereinfache $R_{3}$ .

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

Schritt 4.5

Multiply each element of $R_{3}$ by $- \frac{3}{4}$ to make the entry at $3, 3$ a $1$ .

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Schritt 4.5.1

Multiply each element of $R_{3}$ by $- \frac{3}{4}$ to make the entry at $3, 3$ a $1$ .

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

Schritt 4.5.2

Vereinfache $R_{3}$ .

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

Schritt 4.6

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

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Schritt 4.6.1

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

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

Schritt 4.6.2

Vereinfache $R_{2}$ .

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

Schritt 4.7

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

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Schritt 4.7.1

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

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

Schritt 4.7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{5}{8} & \frac{1}{4} & - \frac{3}{8} \\ 0 & 1 & 0 & - \frac{1}{4} & - \frac{1}{2} & \frac{7}{4} \\ 0 & 0 & 1 & \frac{1}{4} & \frac{1}{2} & - \frac{3}{4} \end{array}]$

Schritt 4.8

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

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Schritt 4.8.1

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

$[\begin{array}{c} 1 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & 0 - \frac{3}{2} \cdot 0 & \frac{5}{8} - \frac{3}{2} (- \frac{1}{4}) & \frac{1}{4} - \frac{3}{2} (- \frac{1}{2}) & - \frac{3}{8} - \frac{3}{2} \cdot \frac{7}{4} \\ 0 & 1 & 0 & - \frac{1}{4} & - \frac{1}{2} & \frac{7}{4} \\ 0 & 0 & 1 & \frac{1}{4} & \frac{1}{2} & - \frac{3}{4} \end{array}]$

Schritt 4.8.2

Vereinfache $R_{1}$ .

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

Schritt 5

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

$[\begin{array}{c} 1 & 1 & - 3 \\ - \frac{1}{4} & - \frac{1}{2} & \frac{7}{4} \\ \frac{1}{4} & \frac{1}{2} & - \frac{3}{4} \end{array}]$