Matemática discreta Ejemplos

$[\begin{array}{c} 1 & 1 & 0 & 0 & 1500 \\ 0 & 0 & 1 & 1 & 1500 \\ 1 & 0 & 1 & 0 & 1000 \\ 0 & 1 & 0 & 1 & 2000 \\ 1 & 4 & 1 & 2 & 8620 \end{array}]$

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 row $1$ 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.

$| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} |$

Paso 1.4

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

$1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} |$

Paso 1.5

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

$| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} |$

Paso 1.6

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

$- 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} |$

Paso 1.7

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

$| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} |$

Paso 1.8

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

$0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} |$

Paso 1.9

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

$| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} |$

Paso 1.10

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

$0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} |$

Paso 1.11

The minor for $a_{15}$ is the determinant with row $1$ and column $5$ deleted.

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

Paso 1.12

Multiply element $a_{15}$ by its cofactor.

$1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 1.13

Add the terms together.

$1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} | + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 2

Multiplica $0$ por $| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 0 & 1000 \\ 0 & 1 & 1 & 2000 \\ 1 & 4 & 2 & 8620 \end{array} |$ .

$1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} | + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 3

Multiplica $0$ por $| \begin{array}{c} 0 & 0 & 1 & 1500 \\ 1 & 0 & 1 & 1000 \\ 0 & 1 & 0 & 2000 \\ 1 & 4 & 1 & 8620 \end{array} |$ .

Paso 4

Evalúa $| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 0 & 1 & 0 & 1000 \\ 1 & 0 & 1 & 2000 \\ 4 & 1 & 2 & 8620 \end{array} |$ .

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

$| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 4.1.4

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

$0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 4.1.5

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

$| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 4.1.6

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

$0 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 4.1.7

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

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

Paso 4.1.8

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

$1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

Paso 4.1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

Paso 4.1.10

Multiply element $a_{41}$ by its cofactor.

$- 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

Paso 4.1.11

Add the terms together.

$1 (0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.2

Multiplica $0$ por $| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 (0 + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.3

Multiplica $0$ por $| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 (0 + 0 + 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4

Evalúa $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$ .

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Paso 4.4.1

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 4.4.1.1

Consider the corresponding sign chart.

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

Paso 4.4.1.2

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

Paso 4.4.1.3

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

$| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

Paso 4.4.1.4

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

$- 1 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

Paso 4.4.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} |$

Paso 4.4.1.6

Multiply element $a_{22}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} |$

Paso 4.4.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

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

Paso 4.4.1.8

Multiply element $a_{23}$ by its cofactor.

$- 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |$

Paso 4.4.1.9

Add the terms together.

$1 (0 + 0 + 1 (- 1 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} | - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.2

Multiplica $0$ por $| \begin{array}{c} 1 & 1500 \\ 1 & 8620 \end{array} |$ .

$1 (0 + 0 + 1 (- 1 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} | + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.3

Evalúa $| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$ .

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Paso 4.4.3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 (- 1 (1 \cdot 8620 - 2 \cdot 1500) + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.3.2

Simplifica el determinante.

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Paso 4.4.3.2.1

Simplifica cada término.

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Paso 4.4.3.2.1.1

Multiplica $8620$ por $1$ .

$1 (0 + 0 + 1 (- 1 (8620 - 2 \cdot 1500) + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.3.2.1.2

Multiplica $- 2$ por $1500$ .

$1 (0 + 0 + 1 (- 1 (8620 - 3000) + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.3.2.2

Resta $3000$ de $8620$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 | \begin{array}{c} 1 & 1 \\ 1 & 2 \end{array} |) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.4

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

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Paso 4.4.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 (1 \cdot 2 - 1 \cdot 1)) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.4.2

Simplifica el determinante.

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Paso 4.4.4.2.1

Simplifica cada término.

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Paso 4.4.4.2.1.1

Multiplica $2$ por $1$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 (2 - 1 \cdot 1)) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.4.2.1.2

Multiplica $- 1$ por $1$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 (2 - 1)) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.4.2.2

Resta $1$ de $2$ .

$1 (0 + 0 + 1 (- 1 \cdot 5620 + 0 - 1000 \cdot 1) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.5

Simplifica el determinante.

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Paso 4.4.5.1

Simplifica cada término.

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Paso 4.4.5.1.1

Multiplica $- 1$ por $5620$ .

$1 (0 + 0 + 1 (- 5620 + 0 - 1000 \cdot 1) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.5.1.2

Multiplica $- 1000$ por $1$ .

$1 (0 + 0 + 1 (- 5620 + 0 - 1000) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.5.2

Suma $- 5620$ y $0$ .

$1 (0 + 0 + 1 (- 5620 - 1000) - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.4.5.3

Resta $1000$ de $- 5620$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5

Evalúa $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$ .

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Paso 4.5.1

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 column $1$ by its cofactor and add.

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Paso 4.5.1.1

Consider the corresponding sign chart.

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

Paso 4.5.1.2

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

Paso 4.5.1.3

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

$| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

Paso 4.5.1.4

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

$1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

Paso 4.5.1.5

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

$| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

Paso 4.5.1.6

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

$- 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

Paso 4.5.1.7

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

$| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

Paso 4.5.1.8

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

$0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

Paso 4.5.1.9

Add the terms together.

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.2

Multiplica $0$ por $| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.3

Evalúa $| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$ .

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Paso 4.5.3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 (0 \cdot 2000 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.3.2

Simplifica el determinante.

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Paso 4.5.3.2.1

Simplifica cada término.

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Paso 4.5.3.2.1.1

Multiplica $0$ por $2000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 (0 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.3.2.1.2

Multiplica $- 1$ por $1000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 (0 - 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.3.2.2

Resta $1000$ de $0$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.4

Evalúa $| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$ .

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Paso 4.5.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 (1 \cdot 2000 - 1 \cdot 1500) + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.4.2

Simplifica el determinante.

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Paso 4.5.4.2.1

Simplifica cada término.

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Paso 4.5.4.2.1.1

Multiplica $2000$ por $1$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 (2000 - 1 \cdot 1500) + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.4.2.1.2

Multiplica $- 1$ por $1500$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 (2000 - 1500) + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.4.2.2

Resta $1500$ de $2000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (1 \cdot - 1000 - 1 \cdot 500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.5

Simplifica el determinante.

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Paso 4.5.5.1

Simplifica cada término.

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Paso 4.5.5.1.1

Multiplica $- 1000$ por $1$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (- 1000 - 1 \cdot 500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.5.1.2

Multiplica $- 1$ por $500$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (- 1000 - 500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.5.2

Resta $500$ de $- 1000$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 (- 1500 + 0)) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.5.5.3

Suma $- 1500$ y $0$ .

$1 (0 + 0 + 1 \cdot - 6620 - 4 \cdot - 1500) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.6

Simplifica el determinante.

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Paso 4.6.1

Simplifica cada término.

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Paso 4.6.1.1

Multiplica $- 6620$ por $1$ .

$1 (0 + 0 - 6620 - 4 \cdot - 1500) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.6.1.2

Multiplica $- 4$ por $- 1500$ .

$1 (0 + 0 - 6620 + 6000) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.6.2

Suma $0$ y $0$ .

$1 (0 - 6620 + 6000) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.6.3

Resta $6620$ de $0$ .

$1 (- 6620 + 6000) - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 4.6.4

Suma $- 6620$ y $6000$ .

$1 \cdot - 620 - 1 | \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} | + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5

Evalúa $| \begin{array}{c} 0 & 1 & 1 & 1500 \\ 1 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \\ 1 & 1 & 2 & 8620 \end{array} |$ .

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

$| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 5.1.4

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

$0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 5.1.5

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

$| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 5.1.6

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

$- 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$

Paso 5.1.7

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

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

Paso 5.1.8

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

$0 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$

Paso 5.1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

Paso 5.1.10

Multiply element $a_{41}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$

Paso 5.1.11

Add the terms together.

$1 \cdot - 620 - 1 (0 | \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.2

Multiplica $0$ por $| \begin{array}{c} 1 & 0 & 1000 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} | - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.3

Multiplica $0$ por $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 1 & 2 & 8620 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} | + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4

Evalúa $| \begin{array}{c} 1 & 1 & 1500 \\ 0 & 1 & 2000 \\ 1 & 2 & 8620 \end{array} |$ .

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Paso 5.4.1

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 column $1$ by its cofactor and add.

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Paso 5.4.1.1

Consider the corresponding sign chart.

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

Paso 5.4.1.2

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

Paso 5.4.1.3

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

$| \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} |$

Paso 5.4.1.4

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

$1 | \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} |$

Paso 5.4.1.5

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

$| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

Paso 5.4.1.6

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

$0 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$

Paso 5.4.1.7

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

$| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

Paso 5.4.1.8

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

$1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

Paso 5.4.1.9

Add the terms together.

$1 \cdot - 620 - 1 (0 - 1 (1 | \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} | + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.2

Multiplica $0$ por $| \begin{array}{c} 1 & 1500 \\ 2 & 8620 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 (1 | \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} | + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.3

Evalúa $| \begin{array}{c} 1 & 2000 \\ 2 & 8620 \end{array} |$ .

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Paso 5.4.3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 (1 (1 \cdot 8620 - 2 \cdot 2000) + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.3.2

Simplifica el determinante.

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Paso 5.4.3.2.1

Simplifica cada término.

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Paso 5.4.3.2.1.1

Multiplica $8620$ por $1$ .

$1 \cdot - 620 - 1 (0 - 1 (1 (8620 - 2 \cdot 2000) + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.3.2.1.2

Multiplica $- 2$ por $2000$ .

$1 \cdot - 620 - 1 (0 - 1 (1 (8620 - 4000) + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.3.2.2

Resta $4000$ de $8620$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.4

Evalúa $| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$ .

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Paso 5.4.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 (1 \cdot 2000 - 1 \cdot 1500)) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.4.2

Simplifica el determinante.

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Paso 5.4.4.2.1

Simplifica cada término.

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Paso 5.4.4.2.1.1

Multiplica $2000$ por $1$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 (2000 - 1 \cdot 1500)) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.4.2.1.2

Multiplica $- 1$ por $1500$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 (2000 - 1500)) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.4.2.2

Resta $1500$ de $2000$ .

$1 \cdot - 620 - 1 (0 - 1 (1 \cdot 4620 + 0 + 1 \cdot 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.5

Simplifica el determinante.

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Paso 5.4.5.1

Simplifica cada término.

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Paso 5.4.5.1.1

Multiplica $4620$ por $1$ .

$1 \cdot - 620 - 1 (0 - 1 (4620 + 0 + 1 \cdot 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.5.1.2

Multiplica $500$ por $1$ .

$1 \cdot - 620 - 1 (0 - 1 (4620 + 0 + 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.5.2

Suma $4620$ y $0$ .

$1 \cdot - 620 - 1 (0 - 1 (4620 + 500) + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.4.5.3

Suma $4620$ y $500$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 | \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5

Evalúa $| \begin{array}{c} 1 & 1 & 1500 \\ 1 & 0 & 1000 \\ 0 & 1 & 2000 \end{array} |$ .

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Paso 5.5.1

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 column $1$ by its cofactor and add.

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Paso 5.5.1.1

Consider the corresponding sign chart.

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

Paso 5.5.1.2

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

Paso 5.5.1.3

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

$| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

Paso 5.5.1.4

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

$1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$

Paso 5.5.1.5

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

$| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

Paso 5.5.1.6

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

$- 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$

Paso 5.5.1.7

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

$| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

Paso 5.5.1.8

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

$0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$

Paso 5.5.1.9

Add the terms together.

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0 | \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.2

Multiplica $0$ por $| \begin{array}{c} 1 & 1500 \\ 0 & 1000 \end{array} |$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 | \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} | - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.3

Evalúa $| \begin{array}{c} 0 & 1000 \\ 1 & 2000 \end{array} |$ .

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Paso 5.5.3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 (0 \cdot 2000 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.3.2

Simplifica el determinante.

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Paso 5.5.3.2.1

Simplifica cada término.

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Paso 5.5.3.2.1.1

Multiplica $0$ por $2000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 (0 - 1 \cdot 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.3.2.1.2

Multiplica $- 1$ por $1000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 (0 - 1000) - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.3.2.2

Resta $1000$ de $0$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 | \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} | + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.4

Evalúa $| \begin{array}{c} 1 & 1500 \\ 1 & 2000 \end{array} |$ .

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Paso 5.5.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 (1 \cdot 2000 - 1 \cdot 1500) + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.4.2

Simplifica el determinante.

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Paso 5.5.4.2.1

Simplifica cada término.

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Paso 5.5.4.2.1.1

Multiplica $2000$ por $1$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 (2000 - 1 \cdot 1500) + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.4.2.1.2

Multiplica $- 1$ por $1500$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 (2000 - 1500) + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.4.2.2

Resta $1500$ de $2000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (1 \cdot - 1000 - 1 \cdot 500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.5

Simplifica el determinante.

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Paso 5.5.5.1

Simplifica cada término.

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Paso 5.5.5.1.1

Multiplica $- 1000$ por $1$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (- 1000 - 1 \cdot 500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.5.1.2

Multiplica $- 1$ por $500$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (- 1000 - 500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.5.2

Resta $500$ de $- 1000$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 (- 1500 + 0)) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.5.5.3

Suma $- 1500$ y $0$ .

$1 \cdot - 620 - 1 (0 - 1 \cdot 5120 + 0 - 1 \cdot - 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.6

Simplifica el determinante.

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Paso 5.6.1

Simplifica cada término.

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Paso 5.6.1.1

Multiplica $- 1$ por $5120$ .

$1 \cdot - 620 - 1 (0 - 5120 + 0 - 1 \cdot - 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.6.1.2

Multiplica $- 1$ por $- 1500$ .

$1 \cdot - 620 - 1 (0 - 5120 + 0 + 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.6.2

Resta $5120$ de $0$ .

$1 \cdot - 620 - 1 (- 5120 + 0 + 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.6.3

Suma $- 5120$ y $0$ .

$1 \cdot - 620 - 1 (- 5120 + 1500) + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 5.6.4

Suma $- 5120$ y $1500$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 | \begin{array}{c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{array} |$

Paso 6

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

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Paso 6.1

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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Paso 6.1.1

Consider the corresponding sign chart.

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

Paso 6.1.2

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

Paso 6.1.3

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

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

Paso 6.1.4

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

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

Paso 6.1.5

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

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

Paso 6.1.6

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

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

Paso 6.1.7

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

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

Paso 6.1.8

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

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

Paso 6.1.9

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

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

Paso 6.1.10

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

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

Paso 6.1.11

Add the terms together.

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 | \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 4 & 1 & 2 \end{array} | + 0 | \begin{array}{c} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 2 \end{array} | + 1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} | - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.2

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

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 | \begin{array}{c} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 2 \end{array} | + 1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} | - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.3

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

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 4 & 2 \end{array} | - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4

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

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Paso 6.4.1

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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Paso 6.4.1.1

Consider the corresponding sign chart.

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

Paso 6.4.1.2

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

Paso 6.4.1.3

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

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

Paso 6.4.1.4

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

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

Paso 6.4.1.5

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

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

Paso 6.4.1.6

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

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

Paso 6.4.1.7

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

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

Paso 6.4.1.8

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

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

Paso 6.4.1.9

Add the terms together.

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 | \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 1 & 2 \end{array} | + 0 | \begin{array}{c} 0 & 1 \\ 1 & 4 \end{array} |) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.2

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

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 | \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} | + 0 + 0 | \begin{array}{c} 0 & 1 \\ 1 & 4 \end{array} |) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.3

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

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 | \begin{array}{c} 1 & 1 \\ 4 & 2 \end{array} | + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.4

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

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Paso 6.4.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 (1 \cdot 2 - 4 \cdot 1) + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.4.2

Simplifica el determinante.

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Paso 6.4.4.2.1

Simplifica cada término.

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Paso 6.4.4.2.1.1

Multiplica $2$ por $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 (2 - 4 \cdot 1) + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.4.2.1.2

Multiplica $- 4$ por $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 (2 - 4) + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.4.2.2

Resta $4$ de $2$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (1 \cdot - 2 + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.5

Simplifica el determinante.

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Paso 6.4.5.1

Multiplica $- 2$ por $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (- 2 + 0 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.5.2

Suma $- 2$ y $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 (- 2 + 0) - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.4.5.3

Suma $- 2$ y $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 | \begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 4 & 1 \end{array} |)$

Paso 6.5

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

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Paso 6.5.1

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 6.5.1.1

Consider the corresponding sign chart.

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

Paso 6.5.1.2

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

Paso 6.5.1.3

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

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

Paso 6.5.1.4

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

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

Paso 6.5.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

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

Paso 6.5.1.6

Multiply element $a_{22}$ by its cofactor.

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

Paso 6.5.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

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

Paso 6.5.1.8

Multiply element $a_{23}$ by its cofactor.

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

Paso 6.5.1.9

Add the terms together.

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 | \begin{array}{c} 0 & 1 \\ 4 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 1 & 4 \end{array} |))$

Paso 6.5.2

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

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 1 & 4 \end{array} |))$

Paso 6.5.3

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

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 | \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array} | + 0))$

Paso 6.5.4

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

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Paso 6.5.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 (1 \cdot 1 - 1 \cdot 1) + 0))$

Paso 6.5.4.2

Simplifica el determinante.

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Paso 6.5.4.2.1

Simplifica cada término.

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Paso 6.5.4.2.1.1

Multiplica $1$ por $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 (1 - 1 \cdot 1) + 0))$

Paso 6.5.4.2.1.2

Multiplica $- 1$ por $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 (1 - 1) + 0))$

Paso 6.5.4.2.2

Resta $1$ de $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 1 \cdot 0 + 0))$

Paso 6.5.5

Simplifica el determinante.

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Paso 6.5.5.1

Multiplica $0$ por $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 0 + 0))$

Paso 6.5.5.2

Suma $0$ y $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 (0 + 0))$

Paso 6.5.5.3

Suma $0$ y $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 + 1 \cdot - 2 - 1 \cdot 0)$

Paso 6.6

Simplifica el determinante.

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Paso 6.6.1

Multiplica $- 2$ por $1$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 + 0 - 2 - 1 \cdot 0)$

Paso 6.6.2

Suma $0$ y $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (0 - 2 - 1 \cdot 0)$

Paso 6.6.3

Resta $2$ de $0$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 (- 2 - 1 \cdot 0)$

Paso 6.6.4

Resta $0$ de $- 2$ .

$1 \cdot - 620 - 1 \cdot - 3620 + 0 + 0 + 1500 \cdot - 2$

Paso 7

Simplifica el determinante.

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Paso 7.1

Simplifica cada término.

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Paso 7.1.1

Multiplica $- 620$ por $1$ .

$- 620 - 1 \cdot - 3620 + 0 + 0 + 1500 \cdot - 2$

Paso 7.1.2

Multiplica $- 1$ por $- 3620$ .

$- 620 + 3620 + 0 + 0 + 1500 \cdot - 2$

Paso 7.1.3

Multiplica $1500$ por $- 2$ .

$- 620 + 3620 + 0 + 0 - 3000$

Paso 7.2

Suma $- 620$ y $3620$ .

$3000 + 0 + 0 - 3000$

Paso 7.3

Suma $3000$ y $0$ .

$3000 + 0 - 3000$

Paso 7.4

Suma $3000$ y $0$ .

$3000 - 3000$

Paso 7.5

Resta $3000$ de $3000$ .

$0$