Matemática discreta Exemplos

$t = [\begin{array}{c} x & 0.4 & 0.3 \\ 0 & y & 0.5 \\ 0.2 & 0.5 & Z \end{array}]$

Etapa 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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Etapa 1.1

Consider the corresponding sign chart.

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

Etapa 1.2

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

Etapa 1.3

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

$| \begin{array}{c} y & 0.5 \\ 0.5 & Z \end{array} |$

Etapa 1.4

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

$x | \begin{array}{c} y & 0.5 \\ 0.5 & Z \end{array} |$

Etapa 1.5

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

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

Etapa 1.6

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

$0 | \begin{array}{c} 0.4 & 0.3 \\ 0.5 & Z \end{array} |$

Etapa 1.7

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

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

Etapa 1.8

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

$0.2 | \begin{array}{c} 0.4 & 0.3 \\ y & 0.5 \end{array} |$

Etapa 1.9

Add the terms together.

$x | \begin{array}{c} y & 0.5 \\ 0.5 & Z \end{array} | + 0 | \begin{array}{c} 0.4 & 0.3 \\ 0.5 & Z \end{array} | + 0.2 | \begin{array}{c} 0.4 & 0.3 \\ y & 0.5 \end{array} |$

Etapa 2

Multiplique $0$ por $| \begin{array}{c} 0.4 & 0.3 \\ 0.5 & Z \end{array} |$ .

$x | \begin{array}{c} y & 0.5 \\ 0.5 & Z \end{array} | + 0 + 0.2 | \begin{array}{c} 0.4 & 0.3 \\ y & 0.5 \end{array} |$

Etapa 3

Avalie $| \begin{array}{c} y & 0.5 \\ 0.5 & Z \end{array} |$ .

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Etapa 3.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x (y Z - 0.5 \cdot 0.5) + 0 + 0.2 | \begin{array}{c} 0.4 & 0.3 \\ y & 0.5 \end{array} |$

Etapa 3.2

Multiplique $- 0.5$ por $0.5$ .

$x (y Z - 0.25) + 0 + 0.2 | \begin{array}{c} 0.4 & 0.3 \\ y & 0.5 \end{array} |$

Etapa 4

Avalie $| \begin{array}{c} 0.4 & 0.3 \\ y & 0.5 \end{array} |$ .

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Etapa 4.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x (y Z - 0.25) + 0 + 0.2 (0.4 \cdot 0.5 - y \cdot 0.3)$

Etapa 4.2

Simplifique cada termo.

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Etapa 4.2.1

Multiplique $0.4$ por $0.5$ .

$x (y Z - 0.25) + 0 + 0.2 (0.2 - y \cdot 0.3)$

Etapa 4.2.2

Multiplique $0.3$ por $- 1$ .

$x (y Z - 0.25) + 0 + 0.2 (0.2 - 0.3 y)$

Etapa 5

Simplifique o determinante.

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Etapa 5.1

Some $x (y Z - 0.25)$ e $0$ .

$x (y Z - 0.25) + 0.2 (0.2 - 0.3 y)$

Etapa 5.2

Simplifique cada termo.

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Etapa 5.2.1

Aplique a propriedade distributiva.

$x (y Z) + x \cdot - 0.25 + 0.2 (0.2 - 0.3 y)$

Etapa 5.2.2

Mova $- 0.25$ para a esquerda de $x$ .

$x y Z - 0.25 \cdot x + 0.2 (0.2 - 0.3 y)$

Etapa 5.2.3

Aplique a propriedade distributiva.

$x y Z - 0.25 x + 0.2 \cdot 0.2 + 0.2 (- 0.3 y)$

Etapa 5.2.4

Multiplique $0.2$ por $0.2$ .

$x y Z - 0.25 x + 0.04 + 0.2 (- 0.3 y)$

Etapa 5.2.5

Multiplique $- 0.3$ por $0.2$ .

$x y Z - 0.25 x + 0.04 - 0.06 y$