Álgebra lineal Ejemplos

$[\begin{array}{c} 6 & 3 & 2 & 4 & 0 \\ 9 & 0 & - 4 & 1 & 0 \\ 8 & - 5 & 6 & 7 & 1 \\ 3 & 0 & 0 & 0 & 0 \\ 4 & 2 & 3 & 2 & 0 \end{array}]$

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 $4$ by its cofactor and add.

Toca para ver más pasos...

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

$| \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} |$

Paso 1.4

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

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} |$

Paso 1.5

The minor for $a_{42}$ is the determinant with row $4$ and column $2$ deleted.

$| \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} |$

Paso 1.6

Multiply element $a_{42}$ by its cofactor.

$0 | \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} |$

Paso 1.7

The minor for $a_{43}$ is the determinant with row $4$ and column $3$ deleted.

$| \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} |$

Paso 1.8

Multiply element $a_{43}$ by its cofactor.

$0 | \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} |$

Paso 1.9

The minor for $a_{44}$ is the determinant with row $4$ and column $4$ deleted.

$| \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} |$

Paso 1.10

Multiply element $a_{44}$ by its cofactor.

$0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} |$

Paso 1.11

The minor for $a_{45}$ is the determinant with row $4$ and column $5$ deleted.

$| \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Paso 1.12

Multiply element $a_{45}$ by its cofactor.

$0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Paso 1.13

Add the terms together.

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Paso 2

Multiplica $0$ por $| \begin{array}{c} 6 & 2 & 4 & 0 \\ 9 & - 4 & 1 & 0 \\ 8 & 6 & 7 & 1 \\ 4 & 3 & 2 & 0 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 | \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Paso 3

Multiplica $0$ por $| \begin{array}{c} 6 & 3 & 4 & 0 \\ 9 & 0 & 1 & 0 \\ 8 & - 5 & 7 & 1 \\ 4 & 2 & 2 & 0 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 + 0 | \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} | + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Paso 4

Multiplica $0$ por $| \begin{array}{c} 6 & 3 & 2 & 0 \\ 9 & 0 & - 4 & 0 \\ 8 & - 5 & 6 & 1 \\ 4 & 2 & 3 & 0 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 + 0 + 0 | \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$

Paso 5

Multiplica $0$ por $| \begin{array}{c} 6 & 3 & 2 & 4 \\ 9 & 0 & - 4 & 1 \\ 8 & - 5 & 6 & 7 \\ 4 & 2 & 3 & 2 \end{array} |$ .

$- 3 | \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} | + 0 + 0 + 0 + 0$

Paso 6

Evalúa $| \begin{array}{c} 3 & 2 & 4 & 0 \\ 0 & - 4 & 1 & 0 \\ - 5 & 6 & 7 & 1 \\ 2 & 3 & 2 & 0 \end{array} |$ .

Toca para ver más pasos...

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 column $4$ by its cofactor and add.

Toca para ver más pasos...

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

$| \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Paso 6.1.4

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

$0 | \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Paso 6.1.5

The minor for $a_{24}$ is the determinant with row $2$ and column $4$ deleted.

$| \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Paso 6.1.6

Multiply element $a_{24}$ by its cofactor.

$0 | \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$

Paso 6.1.7

The minor for $a_{34}$ is the determinant with row $3$ and column $4$ deleted.

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

Paso 6.1.8

Multiply element $a_{34}$ by its cofactor.

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

Paso 6.1.9

The minor for $a_{44}$ is the determinant with row $4$ and column $4$ deleted.

$| \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |$

Paso 6.1.10

Multiply element $a_{44}$ by its cofactor.

$0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |$

Paso 6.1.11

Add the terms together.

$- 3 (0 | \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} | - 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |) + 0 + 0 + 0 + 0$

Paso 6.2

Multiplica $0$ por $| \begin{array}{c} 0 & - 4 & 1 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$ .

$- 3 (0 + 0 | \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} | - 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |) + 0 + 0 + 0 + 0$

Paso 6.3

Multiplica $0$ por $| \begin{array}{c} 3 & 2 & 4 \\ - 5 & 6 & 7 \\ 2 & 3 & 2 \end{array} |$ .

$- 3 (0 + 0 - 1 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ 2 & 3 & 2 \end{array} | + 0 | \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |) + 0 + 0 + 0 + 0$

Paso 6.4

Multiplica $0$ por $| \begin{array}{c} 3 & 2 & 4 \\ 0 & - 4 & 1 \\ - 5 & 6 & 7 \end{array} |$ .

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

Paso 6.5

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

Toca para ver más pasos...

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 column $1$ by its cofactor and add.

Toca para ver más pasos...

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

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

Paso 6.5.1.4

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

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

Paso 6.5.1.5

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

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

Paso 6.5.1.6

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

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

Paso 6.5.1.7

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

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

Paso 6.5.1.8

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

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

Paso 6.5.1.9

Add the terms together.

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

Paso 6.5.2

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

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

Paso 6.5.3

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

Toca para ver más pasos...

Paso 6.5.3.1

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

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

Paso 6.5.3.2

Simplifica el determinante.

Toca para ver más pasos...

Paso 6.5.3.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 6.5.3.2.1.1

Multiplica $- 4$ por $2$ .

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

Paso 6.5.3.2.1.2

Multiplica $- 3$ por $1$ .

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

Paso 6.5.3.2.2

Resta $3$ de $- 8$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 | \begin{array}{c} 2 & 4 \\ - 4 & 1 \end{array} |) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.4

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

Toca para ver más pasos...

Paso 6.5.4.1

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

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 \cdot 1 - (- 4 \cdot 4))) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.4.2

Simplifica el determinante.

Toca para ver más pasos...

Paso 6.5.4.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 6.5.4.2.1.1

Multiplica $2$ por $1$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 - (- 4 \cdot 4))) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.4.2.1.2

Multiplica $- (- 4 \cdot 4)$ .

Toca para ver más pasos...

Paso 6.5.4.2.1.2.1

Multiplica $- 4$ por $4$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 - - 16)) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.4.2.1.2.2

Multiplica $- 1$ por $- 16$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 (2 + 16)) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.4.2.2

Suma $2$ y $16$ .

$- 3 (0 + 0 - 1 (3 \cdot - 11 + 0 + 2 \cdot 18) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.5

Simplifica el determinante.

Toca para ver más pasos...

Paso 6.5.5.1

Simplifica cada término.

Toca para ver más pasos...

Paso 6.5.5.1.1

Multiplica $3$ por $- 11$ .

$- 3 (0 + 0 - 1 (- 33 + 0 + 2 \cdot 18) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.5.1.2

Multiplica $2$ por $18$ .

$- 3 (0 + 0 - 1 (- 33 + 0 + 36) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.5.2

Suma $- 33$ y $0$ .

$- 3 (0 + 0 - 1 (- 33 + 36) + 0) + 0 + 0 + 0 + 0$

Paso 6.5.5.3

Suma $- 33$ y $36$ .

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

Paso 6.6

Simplifica el determinante.

Toca para ver más pasos...

Paso 6.6.1

Multiplica $- 1$ por $3$ .

$- 3 (0 + 0 - 3 + 0) + 0 + 0 + 0 + 0$

Paso 6.6.2

Suma $0$ y $0$ .

$- 3 (0 - 3 + 0) + 0 + 0 + 0 + 0$

Paso 6.6.3

Resta $3$ de $0$ .

$- 3 (- 3 + 0) + 0 + 0 + 0 + 0$

Paso 6.6.4

Suma $- 3$ y $0$ .

$- 3 \cdot - 3 + 0 + 0 + 0 + 0$

Paso 7

Simplifica el determinante.

Toca para ver más pasos...

Paso 7.1

Multiplica $- 3$ por $- 3$ .

$9 + 0 + 0 + 0 + 0$

Paso 7.2

Suma $9$ y $0$ .

$9 + 0 + 0 + 0$

Paso 7.3

Suma $9$ y $0$ .

$9 + 0 + 0$

Paso 7.4

Suma $9$ y $0$ .

$9 + 0$

Paso 7.5

Suma $9$ y $0$ .

$9$