Finite Mathematik Beispiele

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

Schritt 1

Multipliziere $[\begin{array}{c} 5 & - 2 \\ - 4 & 3 \\ 0 & 0 \end{array}] [\begin{array}{c} 1 & - 1 & 4 \\ - 1 & 0 & 3 \end{array}]$ .

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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 $3 \times 2$ and the second matrix is $2 \times 3$ .

Schritt 1.2

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

$[\begin{array}{c} 5 \cdot 1 - 2 \cdot - 1 & 5 \cdot - 1 - 2 \cdot 0 & 5 \cdot 4 - 2 \cdot 3 \\ - 4 \cdot 1 + 3 \cdot - 1 & - 4 \cdot - 1 + 3 \cdot 0 & - 4 \cdot 4 + 3 \cdot 3 \\ 0 \cdot 1 + 0 \cdot - 1 & 0 \cdot - 1 + 0 \cdot 0 & 0 \cdot 4 + 0 \cdot 3 \end{array}]$

Schritt 1.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

$[\begin{array}{c} 7 & - 5 & 14 \\ - 7 & 4 & - 7 \\ 0 & 0 & 0 \end{array}]$

Schritt 2

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 $3$ by its cofactor and add.

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Schritt 2.1

Consider the corresponding sign chart.

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

Schritt 2.2

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

Schritt 2.3

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

$| \begin{array}{c} - 5 & 14 \\ 4 & - 7 \end{array} |$

Schritt 2.4

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

$0 | \begin{array}{c} - 5 & 14 \\ 4 & - 7 \end{array} |$

Schritt 2.5

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 7 & 14 \\ - 7 & - 7 \end{array} |$

Schritt 2.6

Multiply element $a_{32}$ by its cofactor.

$0 | \begin{array}{c} 7 & 14 \\ - 7 & - 7 \end{array} |$

Schritt 2.7

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

$| \begin{array}{c} 7 & - 5 \\ - 7 & 4 \end{array} |$

Schritt 2.8

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

$0 | \begin{array}{c} 7 & - 5 \\ - 7 & 4 \end{array} |$

Schritt 2.9

Add the terms together.

$0 | \begin{array}{c} - 5 & 14 \\ 4 & - 7 \end{array} | + 0 | \begin{array}{c} 7 & 14 \\ - 7 & - 7 \end{array} | + 0 | \begin{array}{c} 7 & - 5 \\ - 7 & 4 \end{array} |$

Schritt 3

Mutltipliziere $0$ mit $| \begin{array}{c} - 5 & 14 \\ 4 & - 7 \end{array} |$ .

$0 + 0 | \begin{array}{c} 7 & 14 \\ - 7 & - 7 \end{array} | + 0 | \begin{array}{c} 7 & - 5 \\ - 7 & 4 \end{array} |$

Schritt 4

Mutltipliziere $0$ mit $| \begin{array}{c} 7 & 14 \\ - 7 & - 7 \end{array} |$ .

$0 + 0 + 0 | \begin{array}{c} 7 & - 5 \\ - 7 & 4 \end{array} |$

Schritt 5

Mutltipliziere $0$ mit $| \begin{array}{c} 7 & - 5 \\ - 7 & 4 \end{array} |$ .

$0 + 0 + 0$

Schritt 6

Vereinfache die Determinante.

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Schritt 6.1

Addiere $0$ und $0$ .

$0 + 0$

Schritt 6.2

Addiere $0$ und $0$ .

$0$