Álgebra linear Exemplos

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

Etapa 1

Subtraia os elementos correspondentes.

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

Etapa 2

Simplify each element.

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Etapa 2.1

Subtraia $4$ de $4$ .

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

Etapa 2.2

Subtraia $0$ de $- 7$ .

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

Etapa 2.3

Subtraia $0$ de $4$ .

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

Etapa 2.4

Subtraia $0$ de $0$ .

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

Etapa 2.5

Subtraia $4$ de $- 3$ .

$[\begin{array}{c} 0 & - 7 & 4 \\ 0 & - 7 & 4 - 0 \\ 0 - 0 & - 8 - 0 & 9 - 4 \end{array}]$

Etapa 2.6

Subtraia $0$ de $4$ .

$[\begin{array}{c} 0 & - 7 & 4 \\ 0 & - 7 & 4 \\ 0 - 0 & - 8 - 0 & 9 - 4 \end{array}]$

Etapa 2.7

Subtraia $0$ de $0$ .

$[\begin{array}{c} 0 & - 7 & 4 \\ 0 & - 7 & 4 \\ 0 & - 8 - 0 & 9 - 4 \end{array}]$

Etapa 2.8

Subtraia $0$ de $- 8$ .

$[\begin{array}{c} 0 & - 7 & 4 \\ 0 & - 7 & 4 \\ 0 & - 8 & 9 - 4 \end{array}]$

Etapa 2.9

Subtraia $4$ de $9$ .

$[\begin{array}{c} 0 & - 7 & 4 \\ 0 & - 7 & 4 \\ 0 & - 8 & 5 \end{array}]$

Etapa 3

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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Etapa 3.1

Consider the corresponding sign chart.

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

Etapa 3.2

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

Etapa 3.3

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

$| \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} |$

Etapa 3.4

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

$0 | \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} |$

Etapa 3.5

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

$| \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} |$

Etapa 3.6

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

$0 | \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} |$

Etapa 3.7

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

$| \begin{array}{c} - 7 & 4 \\ - 7 & 4 \end{array} |$

Etapa 3.8

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

$0 | \begin{array}{c} - 7 & 4 \\ - 7 & 4 \end{array} |$

Etapa 3.9

Add the terms together.

$0 | \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} | + 0 | \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} | + 0 | \begin{array}{c} - 7 & 4 \\ - 7 & 4 \end{array} |$

Etapa 4

Multiplique $0$ por $| \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} |$ .

$0 + 0 | \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} | + 0 | \begin{array}{c} - 7 & 4 \\ - 7 & 4 \end{array} |$

Etapa 5

Multiplique $0$ por $| \begin{array}{c} - 7 & 4 \\ - 8 & 5 \end{array} |$ .

$0 + 0 + 0 | \begin{array}{c} - 7 & 4 \\ - 7 & 4 \end{array} |$

Etapa 6

Multiplique $0$ por $| \begin{array}{c} - 7 & 4 \\ - 7 & 4 \end{array} |$ .

$0 + 0 + 0$

Etapa 7

Simplifique o determinante.

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Etapa 7.1

Some $0$ e $0$ .

$0 + 0$

Etapa 7.2

Some $0$ e $0$ .

$0$