Álgebra lineal Ejemplos

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

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.

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

Paso 1.4

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

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

Paso 1.5

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

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

Paso 1.6

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

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

Paso 1.7

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

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

Paso 1.8

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

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

Paso 1.9

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

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

Paso 1.10

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

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

Paso 1.11

Add the terms together.

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

Paso 2

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

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

Paso 3

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

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

Paso 4

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

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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 $1$ 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.

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

Paso 4.1.4

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

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

Paso 4.1.5

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

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

Paso 4.1.6

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

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

Paso 4.1.7

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

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

Paso 4.1.8

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

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

Paso 4.1.9

Add the terms together.

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

Paso 4.2

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

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

Paso 4.3

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

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Paso 4.3.1

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

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

Paso 4.3.2

Simplifica el determinante.

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Paso 4.3.2.1

Simplifica cada término.

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Paso 4.3.2.1.1

Multiplica $2$ por $4$ .

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

Paso 4.3.2.1.2

Multiplica $- 3$ por $4$ .

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

Paso 4.3.2.2

Resta $12$ de $8$ .

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

Paso 4.4

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

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Paso 4.4.1

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

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

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 $3$ por $1$ .

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

Paso 4.4.2.1.2

Multiplica $- 1$ por $2$ .

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

Paso 4.4.2.2

Resta $2$ de $3$ .

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

Paso 4.5

Simplifica el determinante.

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Paso 4.5.1

Simplifica cada término.

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Paso 4.5.1.1

Multiplica $4$ por $- 4$ .

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

Paso 4.5.1.2

Multiplica $3$ por $1$ .

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

Paso 4.5.2

Suma $- 16$ y $0$ .

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

Paso 4.5.3

Suma $- 16$ y $3$ .

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

Paso 5

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

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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 $1$ 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.

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

Paso 5.1.4

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

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

Paso 5.1.5

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

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

Paso 5.1.6

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

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

Paso 5.1.7

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

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

Paso 5.1.8

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

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

Paso 5.1.9

Add the terms together.

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

Paso 5.2

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

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

Paso 5.3

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

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Paso 5.3.1

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

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

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 $3$ .

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

Paso 5.3.2.1.2

Multiplica $- 2$ por $2$ .

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

Paso 5.3.2.2

Resta $4$ de $6$ .

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

Paso 5.4

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

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Paso 5.4.1

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

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

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 $3$ por $1$ .

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

Paso 5.4.2.1.2

Multiplica $- 1$ por $2$ .

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

Paso 5.4.2.2

Resta $2$ de $3$ .

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

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 $4$ por $2$ .

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

Paso 5.5.1.2

Multiplica $- 3$ por $1$ .

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

Paso 5.5.2

Resta $3$ de $8$ .

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

Paso 5.5.3

Suma $5$ y $0$ .

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

Paso 6

Simplifica el determinante.

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Paso 6.1

Simplifica cada término.

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Paso 6.1.1

Multiplica $- 3$ por $- 13$ .

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

Paso 6.1.2

Multiplica $- 1$ por $5$ .

$0 + 39 + 0 - 5$

Paso 6.2

Suma $0$ y $39$ .

$39 + 0 - 5$

Paso 6.3

Suma $39$ y $0$ .

$39 - 5$

Paso 6.4

Resta $5$ de $39$ .

$34$

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