Álgebra Ejemplos

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

Paso 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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Paso 1.1

Consider the corresponding sign chart.

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

Paso 1.2

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

$-$ position on the sign chart.

Paso 1.3

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

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

Paso 1.4

Multiply element

$a_{11}$ by its cofactor.

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

Paso 1.5

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

Paso 1.6

Multiply element

$a_{21}$ by its cofactor.

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

Paso 1.7

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

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

Paso 1.8

Multiply element

$a_{31}$ by its cofactor.

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

Paso 1.9

The minor for

$a_{41}$ is the determinant with row

$4$ and column

$1$ deleted.

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

Paso 1.10

Multiply element

$a_{41}$ by its cofactor.

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

Paso 1.11

Add the terms together.

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

Paso 2

Multiplica

$0$ por

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

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

Paso 3

Multiplica

$0$ por

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

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

Paso 4

Evalúa

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

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Paso 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

$2$ by its cofactor and add.

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Paso 4.1.1

Consider the corresponding sign chart.

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

Paso 4.1.2

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

$-$ position on the sign chart.

Paso 4.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

Paso 4.1.4

Multiply element

$a_{21}$ by its cofactor.

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

Paso 4.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

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

Paso 4.1.6

Multiply element

$a_{22}$ by its cofactor.

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

Paso 4.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

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

Paso 4.1.8

Multiply element

$a_{23}$ by its cofactor.

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

Paso 4.1.9

Add the terms together.

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

Paso 4.2

Multiplica

$0$ por

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

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

Paso 4.3

Multiplica

$0$ por

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

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

Paso 4.4

Evalúa

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

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Paso 4.4.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

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

Paso 4.4.2

Simplifica el determinante.

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Paso 4.4.2.1

Simplifica cada término.

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Paso 4.4.2.1.1

Multiplica

$1$ por

$1$ .

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

Paso 4.4.2.1.2

Multiplica

$- 2$ por

$1$ .

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

Paso 4.4.2.2

Resta

$2$ de

$1$ .

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

Paso 4.5

Simplifica el determinante.

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Paso 4.5.1

Multiplica

$- 1$ por

$1$ .

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

Paso 4.5.2

Resta

$1$ de

$0$ .

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

Paso 4.5.3

Suma

$- 1$ y

$0$ .

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

Paso 5

Evalúa

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

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Paso 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

$2$ by its cofactor and add.

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Paso 5.1.1

Consider the corresponding sign chart.

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

Paso 5.1.2

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

$-$ position on the sign chart.

Paso 5.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

Paso 5.1.4

Multiply element

$a_{21}$ by its cofactor.

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

Paso 5.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

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

Paso 5.1.6

Multiply element

$a_{22}$ by its cofactor.

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

Paso 5.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

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

Paso 5.1.8

Multiply element

$a_{23}$ by its cofactor.

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

Paso 5.1.9

Add the terms together.

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

Paso 5.2

Multiplica

$0$ por

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

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

Paso 5.3

Evalúa

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

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Paso 5.3.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

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

Paso 5.3.2

Simplifica el determinante.

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Paso 5.3.2.1

Simplifica cada término.

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Paso 5.3.2.1.1

Multiplica

$2$ por

$1$ .

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

Paso 5.3.2.1.2

Multiplica

$- 1$ por

$1$ .

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

Paso 5.3.2.2

Resta

$1$ de

$2$ .

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

Paso 5.4

Evalúa

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

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Paso 5.4.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

$0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 (1 \cdot 1 - 2 \cdot 2)) + 0$

Paso 5.4.2

Simplifica el determinante.

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Paso 5.4.2.1

Simplifica cada término.

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Paso 5.4.2.1.1

Multiplica

$1$ por

$1$ .

$0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 (1 - 2 \cdot 2)) + 0$

Paso 5.4.2.1.2

Multiplica

$- 2$ por

$2$ .

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

Paso 5.4.2.2

Resta

$4$ de

$1$ .

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

Paso 5.5

Simplifica el determinante.

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Paso 5.5.1

Simplifica cada término.

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Paso 5.5.1.1

Multiplica

$- 1$ por

$1$ .

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

Paso 5.5.1.2

Multiplica

$- 2$ por

$- 3$ .

$0 - 1 \cdot - 1 + 1 (- 1 + 0 + 6) + 0$

Paso 5.5.2

Suma

$- 1$ y

$0$ .

$0 - 1 \cdot - 1 + 1 (- 1 + 6) + 0$

Paso 5.5.3

Suma

$- 1$ y

$6$ .

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

Paso 6

Simplifica el determinante.

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Paso 6.1

Simplifica cada término.

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Paso 6.1.1

Multiplica

$- 1$ por

$- 1$ .

$0 + 1 + 1 \cdot 5 + 0$

Paso 6.1.2

Multiplica

$5$ por

$1$ .

$0 + 1 + 5 + 0$

Paso 6.2

Suma

$0$ y

$1$ .

$1 + 5 + 0$

Paso 6.3

Suma

$1$ y

$5$ .

$6 + 0$

Paso 6.4

Suma

$6$ y

$0$ .

$6$

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