Álgebra linear Exemplos

$[\begin{array}{c} a & b & 0 & 0 \\ c & d & 0 & 0 \\ 0 & 0 & a & - b \\ 0 & 0 & c & d \end{array}]$

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 row $1$ 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.

$| \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Etapa 1.4

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

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Etapa 1.5

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

$| \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Etapa 1.6

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

$- b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Etapa 1.7

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

$| \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} |$

Etapa 1.8

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

$0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} |$

Etapa 1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Etapa 1.10

Multiply element $a_{14}$ by its cofactor.

$0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Etapa 1.11

Add the terms together.

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} | + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Etapa 2

Multiplique $0$ por $| \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} |$ .

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Etapa 3

Multiplique $0$ por $| \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$ .

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4

Avalie $| \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$ .

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Etapa 4.1

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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Etapa 4.1.1

Consider the corresponding sign chart.

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

Etapa 4.1.2

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

Etapa 4.1.3

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

$| \begin{array}{c} a & - b \\ c & d \end{array} |$

Etapa 4.1.4

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

$d | \begin{array}{c} a & - b \\ c & d \end{array} |$

Etapa 4.1.5

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

$| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Etapa 4.1.6

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

$0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Etapa 4.1.7

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

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

Etapa 4.1.8

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

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

Etapa 4.1.9

Add the terms together.

$a (d | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} | + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.2

Multiplique $0$ por $| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$ .

$a (d | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.3

Multiplique $0$ por $| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$ .

$a (d | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.4

Avalie $| \begin{array}{c} a & - b \\ c & d \end{array} |$ .

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Etapa 4.4.1

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

$a (d (a d - c (- b)) + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.4.2

Simplifique cada termo.

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Etapa 4.4.2.1

Reescreva usando a propriedade comutativa da multiplicação.

$a (d (a d - 1 \cdot - 1 c b) + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.4.2.2

Multiplique $- 1$ por $- 1$ .

$a (d (a d + 1 c b) + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.4.2.3

Multiplique $c$ por $1$ .

$a (d (a d + c b) + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.5

Simplifique o determinante.

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Etapa 4.5.1

Combine os termos opostos em $d (a d + c b) + 0 + 0$ .

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Etapa 4.5.1.1

Some $d (a d + c b)$ e $0$ .

$a (d (a d + c b) + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.5.1.2

Some $d (a d + c b)$ e $0$ .

$a (d (a d + c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.5.2

Aplique a propriedade distributiva.

$a (d (a d) + d (c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.5.3

Multiplique $d$ por $d$ somando os expoentes.

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Etapa 4.5.3.1

Mova $d$ .

$a (d \cdot d a + d (c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 4.5.3.2

Multiplique $d$ por $d$ .

$a (d^{2} a + d (c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

$a (d^{2} a + d c b) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Etapa 5

Avalie $| \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$ .

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Etapa 5.1

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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Etapa 5.1.1

Consider the corresponding sign chart.

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

Etapa 5.1.2

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

Etapa 5.1.3

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

$| \begin{array}{c} a & - b \\ c & d \end{array} |$

Etapa 5.1.4

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

$c | \begin{array}{c} a & - b \\ c & d \end{array} |$

Etapa 5.1.5

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

$| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Etapa 5.1.6

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

$0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Etapa 5.1.7

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

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

Etapa 5.1.8

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

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

Etapa 5.1.9

Add the terms together.

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} | + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) + 0 + 0$

Etapa 5.2

Multiplique $0$ por $| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$ .

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) + 0 + 0$

Etapa 5.3

Multiplique $0$ por $| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$ .

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0) + 0 + 0$

Etapa 5.4

Avalie $| \begin{array}{c} a & - b \\ c & d \end{array} |$ .

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Etapa 5.4.1

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

$a (d^{2} a + d c b) - b (c (a d - c (- b)) + 0 + 0) + 0 + 0$

Etapa 5.4.2

Simplifique cada termo.

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Etapa 5.4.2.1

Reescreva usando a propriedade comutativa da multiplicação.

$a (d^{2} a + d c b) - b (c (a d - 1 \cdot - 1 c b) + 0 + 0) + 0 + 0$

Etapa 5.4.2.2

Multiplique $- 1$ por $- 1$ .

$a (d^{2} a + d c b) - b (c (a d + 1 c b) + 0 + 0) + 0 + 0$

Etapa 5.4.2.3

Multiplique $c$ por $1$ .

$a (d^{2} a + d c b) - b (c (a d + c b) + 0 + 0) + 0 + 0$

Etapa 5.5

Simplifique o determinante.

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Etapa 5.5.1

Combine os termos opostos em $c (a d + c b) + 0 + 0$ .

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Etapa 5.5.1.1

Some $c (a d + c b)$ e $0$ .

$a (d^{2} a + d c b) - b (c (a d + c b) + 0) + 0 + 0$

Etapa 5.5.1.2

Some $c (a d + c b)$ e $0$ .

$a (d^{2} a + d c b) - b (c (a d + c b)) + 0 + 0$

Etapa 5.5.2

Aplique a propriedade distributiva.

$a (d^{2} a + d c b) - b (c (a d) + c (c b)) + 0 + 0$

Etapa 5.5.3

Multiplique $c$ por $c$ somando os expoentes.

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Etapa 5.5.3.1

Mova $c$ .

$a (d^{2} a + d c b) - b (c a d + c \cdot c b) + 0 + 0$

Etapa 5.5.3.2

Multiplique $c$ por $c$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0 + 0$

Etapa 6

Simplifique o determinante.

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Etapa 6.1

Combine os termos opostos em $a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0 + 0$ .

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Etapa 6.1.1

Some $a (d^{2} a + d c b) - b (c a d + c^{2} b)$ e $0$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0$

Etapa 6.1.2

Some $a (d^{2} a + d c b) - b (c a d + c^{2} b)$ e $0$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b)$

Etapa 6.2

Simplifique cada termo.

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Etapa 6.2.1

Aplique a propriedade distributiva.

$a (d^{2} a) + a (d c b) - b (c a d + c^{2} b)$

Etapa 6.2.2

Multiplique $a$ por $a$ somando os expoentes.

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Etapa 6.2.2.1

Mova $a$ .

$a \cdot a d^{2} + a (d c b) - b (c a d + c^{2} b)$

Etapa 6.2.2.2

Multiplique $a$ por $a$ .

$a^{2} d^{2} + a (d c b) - b (c a d + c^{2} b)$

Etapa 6.2.3

Aplique a propriedade distributiva.

$a^{2} d^{2} + a d c b - b (c a d) - b (c^{2} b)$

Etapa 6.2.4

Multiplique $b$ por $b$ somando os expoentes.

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Etapa 6.2.4.1

Mova $b$ .

$a^{2} d^{2} + a d c b - b c a d - (b \cdot b) c^{2}$

Etapa 6.2.4.2

Multiplique $b$ por $b$ .

$a^{2} d^{2} + a d c b - b c a d - b^{2} c^{2}$

Etapa 6.3

Combine os termos opostos em $a^{2} d^{2} + a d c b - b c a d - b^{2} c^{2}$ .

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Etapa 6.3.1

Reorganize os fatores nos termos $a d c b$ e $- b c a d$ .

$a^{2} d^{2} + a b c d - a b c d - b^{2} c^{2}$

Etapa 6.3.2

Subtraia $a b c d$ de $a b c d$ .

$a^{2} d^{2} + 0 - b^{2} c^{2}$

Etapa 6.3.3

Some $a^{2} d^{2}$ e $0$ .

$a^{2} d^{2} - b^{2} c^{2}$