Exemplos

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & 3 & 0 & 1 4 & 3 & 0 & 3 1 & 2 & 2 & 4 1 & 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Etapa 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.

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Etapa 1.1

Consider the corresponding sign chart.

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

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

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}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 3 & 0 & 3 2 & 2 & 4 2 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

0 ∣ ∣ ∣ ∣ \begin{matrix} 3 & 0 & 3 2 & 2 & 4 2 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 4 & 0 & 3 1 & 2 & 4 1 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- 3 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 0 & 3 1 & 2 & 4 1 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 3 1 & 2 & 4 1 & 2 & 4 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

0 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 3 1 & 2 & 4 1 & 2 & 4 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.9

The minor for

a_{14}

$a_{14}$ is the determinant with row

1

$1$ and column

4

$4$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 0 1 & 2 & 2 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.10

Multiply element

a_{14}

$a_{14}$ by its cofactor.

- 1 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 0 1 & 2 & 2 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣

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

Etapa 1.11

Add the terms together.

0 ∣ ∣ ∣ ∣ \begin{matrix} 3 & 0 & 3 2 & 2 & 4 2 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣ - 3 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 0 & 3 1 & 2 & 4 1 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣ + 0 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 3 1 & 2 & 4 1 & 2 & 4 \end{matrix} ∣ ∣ ∣ ∣ - 1 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 0 1 & 2 & 2 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣

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

0 ∣ ∣ ∣ ∣ \begin{matrix} 3 & 0 & 3 2 & 2 & 4 2 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣ - 3 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 0 & 3 1 & 2 & 4 1 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣ + 0 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 3 1 & 2 & 4 1 & 2 & 4 \end{matrix} ∣ ∣ ∣ ∣ - 1 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 3 & 0 1 & 2 & 2 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣

Etapa 2

Multiplique

$0$ por

$| \begin{array}{c} 3 & 0 & 3 \\ 2 & 2 & 4 \\ 2 & 3 & 4 \end{array} |$ .

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

Etapa 3

Multiplique

$0$ por

$| \begin{array}{c} 4 & 3 & 3 \\ 1 & 2 & 4 \\ 1 & 2 & 4 \end{array} |$ .

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

Etapa 4

Avalie

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

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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.

$| \begin{array}{c} 2 & 4 \\ 3 & 4 \end{array} |$

Etapa 4.1.4

Multiply element

$a_{11}$ by its cofactor.

$4 | \begin{array}{c} 2 & 4 \\ 3 & 4 \end{array} |$

Etapa 4.1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

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

Etapa 4.1.6

Multiply element

$a_{12}$ by its cofactor.

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

Etapa 4.1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

$| \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |$

Etapa 4.1.8

Multiply element

$a_{13}$ by its cofactor.

$3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |$

Etapa 4.1.9

Add the terms together.

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

Etapa 4.2

Multiplique

$0$ por

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

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

Etapa 4.3

Avalie

$| \begin{array}{c} 2 & 4 \\ 3 & 4 \end{array} |$ .

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Etapa 4.3.1

O determinante de uma matriz

$2 \times 2$ pode ser encontrado ao usar a fórmula

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

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

Etapa 4.3.2

Simplifique o determinante.

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Etapa 4.3.2.1

Simplifique cada termo.

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Etapa 4.3.2.1.1

Multiplique

$2$ por

$4$ .

$0 - 3 (4 (8 - 3 \cdot 4) + 0 + 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |) + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Etapa 4.3.2.1.2

Multiplique

$- 3$ por

$4$ .

$0 - 3 (4 (8 - 12) + 0 + 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |) + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Etapa 4.3.2.2

Subtraia

$12$ de

$8$ .

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

Etapa 4.4

Avalie

$| \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |$ .

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Etapa 4.4.1

O determinante de uma matriz

$2 \times 2$ pode ser encontrado ao usar a fórmula

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

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

Etapa 4.4.2

Simplifique o determinante.

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Etapa 4.4.2.1

Simplifique cada termo.

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Etapa 4.4.2.1.1

Multiplique

$3$ por

$1$ .

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

Etapa 4.4.2.1.2

Multiplique

$- 1$ por

$2$ .

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

Etapa 4.4.2.2

Subtraia

$2$ de

$3$ .

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

Etapa 4.5

Simplifique o determinante.

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Etapa 4.5.1

Simplifique cada termo.

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Etapa 4.5.1.1

Multiplique

$4$ por

$- 4$ .

$0 - 3 (- 16 + 0 + 3 \cdot 1) + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Etapa 4.5.1.2

Multiplique

$3$ por

$1$ .

$0 - 3 (- 16 + 0 + 3) + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Etapa 4.5.2

Some

$- 16$ e

$0$ .

$0 - 3 (- 16 + 3) + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Etapa 4.5.3

Some

$- 16$ e

$3$ .

$0 - 3 \cdot - 13 + 0 - 1 | \begin{array}{c} 4 & 3 & 0 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} |$

Etapa 5

Avalie

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

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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.

$| \begin{array}{c} 2 & 2 \\ 2 & 3 \end{array} |$

Etapa 5.1.4

Multiply element

$a_{11}$ by its cofactor.

$4 | \begin{array}{c} 2 & 2 \\ 2 & 3 \end{array} |$

Etapa 5.1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

$| \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |$

Etapa 5.1.6

Multiply element

$a_{12}$ by its cofactor.

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

Etapa 5.1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

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

Etapa 5.1.8

Multiply element

$a_{13}$ by its cofactor.

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

Etapa 5.1.9

Add the terms together.

$0 - 3 \cdot - 13 + 0 - 1 (4 | \begin{array}{c} 2 & 2 \\ 2 & 3 \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 1 & 2 \end{array} |)$

Etapa 5.2

Multiplique

$0$ por

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

$0 - 3 \cdot - 13 + 0 - 1 (4 | \begin{array}{c} 2 & 2 \\ 2 & 3 \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Etapa 5.3

Avalie

$| \begin{array}{c} 2 & 2 \\ 2 & 3 \end{array} |$ .

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Etapa 5.3.1

O determinante de uma matriz

$2 \times 2$ pode ser encontrado ao usar a fórmula

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

$0 - 3 \cdot - 13 + 0 - 1 (4 (2 \cdot 3 - 2 \cdot 2) - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Etapa 5.3.2

Simplifique o determinante.

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Etapa 5.3.2.1

Simplifique cada termo.

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Etapa 5.3.2.1.1

Multiplique

$2$ por

$3$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 (6 - 2 \cdot 2) - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Etapa 5.3.2.1.2

Multiplique

$- 2$ por

$2$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 (6 - 4) - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Etapa 5.3.2.2

Subtraia

$4$ de

$6$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 | \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} | + 0)$

Etapa 5.4

Avalie

$| \begin{array}{c} 1 & 2 \\ 1 & 3 \end{array} |$ .

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Etapa 5.4.1

O determinante de uma matriz

$2 \times 2$ pode ser encontrado ao usar a fórmula

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

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 (1 \cdot 3 - 1 \cdot 2) + 0)$

Etapa 5.4.2

Simplifique o determinante.

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Etapa 5.4.2.1

Simplifique cada termo.

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Etapa 5.4.2.1.1

Multiplique

$3$ por

$1$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 (3 - 1 \cdot 2) + 0)$

Etapa 5.4.2.1.2

Multiplique

$- 1$ por

$2$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 (3 - 2) + 0)$

Etapa 5.4.2.2

Subtraia

$2$ de

$3$ .

$0 - 3 \cdot - 13 + 0 - 1 (4 \cdot 2 - 3 \cdot 1 + 0)$

Etapa 5.5

Simplifique o determinante.

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Etapa 5.5.1

Simplifique cada termo.

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Etapa 5.5.1.1

Multiplique

$4$ por

$2$ .

$0 - 3 \cdot - 13 + 0 - 1 (8 - 3 \cdot 1 + 0)$

Etapa 5.5.1.2

Multiplique

$- 3$ por

$1$ .

$0 - 3 \cdot - 13 + 0 - 1 (8 - 3 + 0)$

Etapa 5.5.2

Subtraia

$3$ de

$8$ .

$0 - 3 \cdot - 13 + 0 - 1 (5 + 0)$

Etapa 5.5.3

Some

$5$ e

$0$ .

$0 - 3 \cdot - 13 + 0 - 1 \cdot 5$

Etapa 6

Simplifique o determinante.

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Etapa 6.1

Simplifique cada termo.

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Etapa 6.1.1

Multiplique

$- 3$ por

$- 13$ .

$0 + 39 + 0 - 1 \cdot 5$

Etapa 6.1.2

Multiplique

$- 1$ por

$5$ .

$0 + 39 + 0 - 5$

Etapa 6.2

Some

$0$ e

$39$ .

$39 + 0 - 5$

Etapa 6.3

Some

$39$ e

$0$ .

$39 - 5$

Etapa 6.4

Subtraia

$5$ de

$39$ .

$34$

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