Matemática discreta Ejemplos

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

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

Paso 1

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in column

1

$1$ by its cofactor and add.

Toca para ver más pasos...

Paso 1.1

Consider the corresponding sign chart.

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

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

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}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

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

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

Paso 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

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

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

Paso 1.5

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

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

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

Paso 1.6

Multiply element

a_{21}

$a_{21}$ by its cofactor.

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

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

Paso 1.7

The minor for

a_{31}

$a_{31}$ is the determinant with row

3

$3$ and column

1

$1$ deleted.

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

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

Paso 1.8

Multiply element

a_{31}

$a_{31}$ by its cofactor.

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

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

Paso 1.9

The minor for

a_{41}

$a_{41}$ is the determinant with row

4

$4$ and column

1

$1$ deleted.

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

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

Paso 1.10

Multiply element

a_{41}

$a_{41}$ by its cofactor.

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

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

Paso 1.11

Add the terms together.

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

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

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

Paso 2

Multiplica

0

$0$ por

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

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

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

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

Paso 3

Multiplica

0

$0$ por

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

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

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

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

Paso 4

Evalúa

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

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

Toca para ver más pasos...

Paso 4.1

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

2

$2$ by its cofactor and add.

Toca para ver más pasos...

Paso 4.1.1

Consider the corresponding sign chart.

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

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

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}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

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

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

Paso 4.1.4

Multiply element

a_{21}

$a_{21}$ by its cofactor.

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

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

Paso 4.1.5

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

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

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

Paso 4.1.6

Multiply element

a_{22}

$a_{22}$ by its cofactor.

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

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

Paso 4.1.7

The minor for

a_{23}

$a_{23}$ is the determinant with row

2

$2$ and column

3

$3$ deleted.

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

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

Paso 4.1.8

Multiply element

a_{23}

$a_{23}$ by its cofactor.

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

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

Paso 4.1.9

Add the terms together.

0 - 1 (0 ∣ ∣ ∣ \begin{matrix} 2 & 1 1 & 1 \end{matrix} ∣ ∣ ∣ + 1 ∣ ∣ ∣ \begin{matrix} 1 & 1 2 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

0 - 1 (0 ∣ ∣ ∣ \begin{matrix} 2 & 1 1 & 1 \end{matrix} ∣ ∣ ∣ + 1 ∣ ∣ ∣ \begin{matrix} 1 & 1 2 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

Paso 4.2

Multiplica

0

$0$ por

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

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

0 - 1 (0 + 1 ∣ ∣ ∣ \begin{matrix} 1 & 1 2 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

Paso 4.3

Multiplica

0

$0$ por

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

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

0 - 1 (0 + 1 ∣ ∣ ∣ \begin{matrix} 1 & 1 2 & 1 \end{matrix} ∣ ∣ ∣ + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

Paso 4.4

Evalúa

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

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

Toca para ver más pasos...

Paso 4.4.1

El determinante de una matriz

2 \times 2

$2 \times 2$ puede obtenerse usando la fórmula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

0 - 1 (0 + 1 (1 \cdot 1 - 2 \cdot 1) + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

Paso 4.4.2

Simplifica el determinante.

Toca para ver más pasos...

Paso 4.4.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 4.4.2.1.1

Multiplica

1

$1$ por

1

$1$ .

0 - 1 (0 + 1 (1 - 2 \cdot 1) + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

Paso 4.4.2.1.2

Multiplica

- 2

$- 2$ por

1

$1$ .

0 - 1 (0 + 1 (1 - 2) + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

0 - 1 (0 + 1 (1 - 2) + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

Paso 4.4.2.2

Resta

2

$2$ de

1

$1$ .

0 - 1 (0 + 1 \cdot - 1 + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

0 - 1 (0 + 1 \cdot - 1 + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

0 - 1 (0 + 1 \cdot - 1 + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

Paso 4.5

Simplifica el determinante.

Toca para ver más pasos...

Paso 4.5.1

Multiplica

- 1

$- 1$ por

1

$1$ .

0 - 1 (0 - 1 + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0

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

Paso 4.5.2

Resta

1

$1$ de

0

$0$ .

0 - 1 (- 1 + 0) + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 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

$- 1$ y

0

$0$ .

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

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

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

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

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

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

Toca para ver más pasos...

Paso 5.1

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

2

$2$ by its cofactor and add.

Toca para ver más pasos...

Paso 5.1.1

Consider the corresponding sign chart.

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

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

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}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

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

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

Paso 5.1.4

Multiply element

a_{21}

$a_{21}$ by its cofactor.

- 1 ∣ ∣ ∣ \begin{matrix} 2 & 1 1 & 1 \end{matrix} ∣ ∣ ∣

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

Paso 5.1.5

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

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

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

Paso 5.1.6

Multiply element

a_{22}

$a_{22}$ by its cofactor.

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

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

Paso 5.1.7

The minor for

a_{23}

$a_{23}$ is the determinant with row

2

$2$ and column

3

$3$ deleted.

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

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

Paso 5.1.8

Multiply element

a_{23}

$a_{23}$ by its cofactor.

- 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣

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

Paso 5.1.9

Add the terms together.

0 - 1 \cdot - 1 + 1 (- 1 ∣ ∣ ∣ \begin{matrix} 2 & 1 1 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 1 & 1 2 & 1 \end{matrix} ∣ ∣ ∣ - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

0 - 1 \cdot - 1 + 1 (- 1 ∣ ∣ ∣ \begin{matrix} 2 & 1 1 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 1 & 1 2 & 1 \end{matrix} ∣ ∣ ∣ - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

Paso 5.2

Multiplica

0

$0$ por

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

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

0 - 1 \cdot - 1 + 1 (- 1 ∣ ∣ ∣ \begin{matrix} 2 & 1 1 & 1 \end{matrix} ∣ ∣ ∣ + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

Paso 5.3

Evalúa

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

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

Toca para ver más pasos...

Paso 5.3.1

El determinante de una matriz

2 \times 2

$2 \times 2$ puede obtenerse usando la fórmula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

0 - 1 \cdot - 1 + 1 (- 1 (2 \cdot 1 - 1 \cdot 1) + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

Paso 5.3.2

Simplifica el determinante.

Toca para ver más pasos...

Paso 5.3.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 5.3.2.1.1

Multiplica

2

$2$ por

1

$1$ .

0 - 1 \cdot - 1 + 1 (- 1 (2 - 1 \cdot 1) + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

Paso 5.3.2.1.2

Multiplica

- 1

$- 1$ por

1

$1$ .

0 - 1 \cdot - 1 + 1 (- 1 (2 - 1) + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

0 - 1 \cdot - 1 + 1 (- 1 (2 - 1) + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

Paso 5.3.2.2

Resta

1

$1$ de

2

$2$ .

0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 ∣ ∣ ∣ \begin{matrix} 1 & 2 2 & 1 \end{matrix} ∣ ∣ ∣) + 0

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

Paso 5.4

Evalúa

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

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

Toca para ver más pasos...

Paso 5.4.1

El determinante de una matriz

2 \times 2

$2 \times 2$ puede obtenerse usando la fórmula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

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

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

Paso 5.4.2

Simplifica el determinante.

Toca para ver más pasos...

Paso 5.4.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 5.4.2.1.1

Multiplica

1

$1$ por

1

$1$ .

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

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

Paso 5.4.2.1.2

Multiplica

- 2

$- 2$ por

2

$2$ .

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

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

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

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

Paso 5.4.2.2

Resta

4

$4$ de

1

$1$ .

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

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

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

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

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

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

Paso 5.5

Simplifica el determinante.

Toca para ver más pasos...

Paso 5.5.1

Simplifica cada término.

Toca para ver más pasos...

Paso 5.5.1.1

Multiplica

- 1

$- 1$ por

1

$1$ .

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

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

Paso 5.5.1.2

Multiplica

- 2

$- 2$ por

- 3

$- 3$ .

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

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

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

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

Paso 5.5.2

Suma

- 1

$- 1$ y

0

$0$ .

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

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

Paso 5.5.3

Suma

- 1

$- 1$ y

6

$6$ .

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

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

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

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

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

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

Paso 6

Simplifica el determinante.

Toca para ver más pasos...

Paso 6.1

Simplifica cada término.

Toca para ver más pasos...

Paso 6.1.1

Multiplica

- 1

$- 1$ por

- 1

$- 1$ .

0 + 1 + 1 \cdot 5 + 0

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

Paso 6.1.2

Multiplica

5

$5$ por

1

$1$ .

0 + 1 + 5 + 0

$0 + 1 + 5 + 0$

0 + 1 + 5 + 0

$0 + 1 + 5 + 0$

Paso 6.2

Suma

0

$0$ y

1

$1$ .

1 + 5 + 0

$1 + 5 + 0$

Paso 6.3

Suma

1

$1$ y

5

$5$ .

6 + 0

$6 + 0$

Paso 6.4

Suma

6

$6$ y

0

$0$ .

6

$6$

6

$6$

Ingresa TU problema