Lineare Algebra Beispiele

⎡ ⎢ ⎣ \begin{matrix} 1 & 7 & 1 2 & 9 & 0 1 & 8 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 4.5 & 0.5 & - 4.5 - 1 & 0 & 1 3.5 & - 0.5 & - 2.5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 7 & 1 \\ 2 & 9 & 0 \\ 1 & 8 & 1 \end{array}] [\begin{array}{c} 4.5 & 0.5 & - 4.5 \\ - 1 & 0 & 1 \\ 3.5 & - 0.5 & - 2.5 \end{array}]$

Schritt 1

Multipliziere

⎡ ⎢ ⎣ \begin{matrix} 1 & 7 & 1 2 & 9 & 0 1 & 8 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 4.5 & 0.5 & - 4.5 - 1 & 0 & 1 3.5 & - 0.5 & - 2.5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 7 & 1 \\ 2 & 9 & 0 \\ 1 & 8 & 1 \end{array}] [\begin{array}{c} 4.5 & 0.5 & - 4.5 \\ - 1 & 0 & 1 \\ 3.5 & - 0.5 & - 2.5 \end{array}]$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is

3 \times 3

$3 \times 3$ and the second matrix is

3 \times 3

$3 \times 3$ .

Schritt 1.2

Multipliziere jede Zeile in der ersten Matrix mit jeder Spalte in der zweiten Matrix.

⎡ ⎢ ⎣ \begin{matrix} 1 \cdot 4.5 + 7 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 7 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 7 \cdot 1 + 1 \cdot - 2.5 2 \cdot 4.5 + 9 \cdot - 1 + 0 \cdot 3.5 & 2 \cdot 0.5 + 9 \cdot 0 + 0 \cdot - 0.5 & 2 \cdot - 4.5 + 9 \cdot 1 + 0 \cdot - 2.5 1 \cdot 4.5 + 8 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 8 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 8 \cdot 1 + 1 \cdot - 2.5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 \cdot 4.5 + 7 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 7 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 7 \cdot 1 + 1 \cdot - 2.5 \\ 2 \cdot 4.5 + 9 \cdot - 1 + 0 \cdot 3.5 & 2 \cdot 0.5 + 9 \cdot 0 + 0 \cdot - 0.5 & 2 \cdot - 4.5 + 9 \cdot 1 + 0 \cdot - 2.5 \\ 1 \cdot 4.5 + 8 \cdot - 1 + 1 \cdot 3.5 & 1 \cdot 0.5 + 8 \cdot 0 + 1 \cdot - 0.5 & 1 \cdot - 4.5 + 8 \cdot 1 + 1 \cdot - 2.5 \end{array}]$

Schritt 1.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

Schritt 2

Find the determinant.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1

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 row

1

$1$ by its cofactor and add.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.1

Consider the corresponding sign chart.

∣ ∣ ∣ ∣ \begin{matrix} + & - & + - & + & - + & - & + \end{matrix} ∣ ∣ ∣ ∣

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

Schritt 2.1.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

Schritt 2.1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 1 & 0 0 & 1 \end{matrix} ∣ ∣ ∣

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

Schritt 2.1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

1 ∣ ∣ ∣ \begin{matrix} 1 & 0 0 & 1 \end{matrix} ∣ ∣ ∣

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

Schritt 2.1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 0 & 0 0 & 1 \end{matrix} ∣ ∣ ∣

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

Schritt 2.1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

0 ∣ ∣ ∣ \begin{matrix} 0 & 0 0 & 1 \end{matrix} ∣ ∣ ∣

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

Schritt 2.1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

∣ ∣ ∣ \begin{matrix} 0 & 1 0 & 0 \end{matrix} ∣ ∣ ∣

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

Schritt 2.1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

0 ∣ ∣ ∣ \begin{matrix} 0 & 1 0 & 0 \end{matrix} ∣ ∣ ∣

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

Schritt 2.1.9

Add the terms together.

1 ∣ ∣ ∣ \begin{matrix} 1 & 0 0 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 0 & 0 0 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 0 & 1 0 & 0 \end{matrix} ∣ ∣ ∣

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

Schritt 2.2

Mutltipliziere

$0$ mit

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

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

Schritt 2.3

Mutltipliziere

$0$ mit

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

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

Schritt 2.4

Berechne

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

Tippen, um mehr Schritte zu sehen ...

Schritt 2.4.1

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.

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

Schritt 2.4.2

Vereinfache die Determinante.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.4.2.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.4.2.1.1

Mutltipliziere

$1$ mit

$1$ .

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

Schritt 2.4.2.1.2

Mutltipliziere

$0$ mit

$0$ .

$1 (1 + 0) + 0 + 0$

Schritt 2.4.2.2

Addiere

$1$ und

$0$ .

$1 \cdot 1 + 0 + 0$

Schritt 2.5

Vereinfache die Determinante.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.5.1

Mutltipliziere

$1$ mit

$1$ .

$1 + 0 + 0$

Schritt 2.5.2

Addiere

$1$ und

$0$ .

$1 + 0$

Schritt 2.5.3

Addiere

$1$ und

$0$ .

$1$

Schritt 3

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

Schritt 4

Set up a

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

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

Schritt 5

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

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