Álgebra linear Exemplos

$[\begin{array}{c} 4 & 10 & 30 \\ 10 & 30 & 100 \\ 30 & 100 & 354 \end{array}]$

Etapa 1

Find the determinant.

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Etapa 1.1

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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Etapa 1.1.1

Consider the corresponding sign chart.

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

Etapa 1.1.2

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

Etapa 1.1.3

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

$| \begin{array}{c} 30 & 100 \\ 100 & 354 \end{array} |$

Etapa 1.1.4

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

$4 | \begin{array}{c} 30 & 100 \\ 100 & 354 \end{array} |$

Etapa 1.1.5

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

$| \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} |$

Etapa 1.1.6

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

$- 10 | \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} |$

Etapa 1.1.7

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

$| \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.1.8

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

$30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.1.9

Add the terms together.

$4 | \begin{array}{c} 30 & 100 \\ 100 & 354 \end{array} | - 10 | \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} | + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.2

Avalie $| \begin{array}{c} 30 & 100 \\ 100 & 354 \end{array} |$ .

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Etapa 1.2.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$4 (30 \cdot 354 - 100 \cdot 100) - 10 | \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} | + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.2.2

Simplifique o determinante.

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Etapa 1.2.2.1

Simplifique cada termo.

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Etapa 1.2.2.1.1

Multiplique $30$ por $354$ .

$4 (10620 - 100 \cdot 100) - 10 | \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} | + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.2.2.1.2

Multiplique $- 100$ por $100$ .

$4 (10620 - 10000) - 10 | \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} | + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.2.2.2

Subtraia $10000$ de $10620$ .

$4 \cdot 620 - 10 | \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} | + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.3

Avalie $| \begin{array}{c} 10 & 100 \\ 30 & 354 \end{array} |$ .

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Etapa 1.3.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$4 \cdot 620 - 10 (10 \cdot 354 - 30 \cdot 100) + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.3.2

Simplifique o determinante.

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Etapa 1.3.2.1

Simplifique cada termo.

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Etapa 1.3.2.1.1

Multiplique $10$ por $354$ .

$4 \cdot 620 - 10 (3540 - 30 \cdot 100) + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.3.2.1.2

Multiplique $- 30$ por $100$ .

$4 \cdot 620 - 10 (3540 - 3000) + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.3.2.2

Subtraia $3000$ de $3540$ .

$4 \cdot 620 - 10 \cdot 540 + 30 | \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$

Etapa 1.4

Avalie $| \begin{array}{c} 10 & 30 \\ 30 & 100 \end{array} |$ .

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Etapa 1.4.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$4 \cdot 620 - 10 \cdot 540 + 30 (10 \cdot 100 - 30 \cdot 30)$

Etapa 1.4.2

Simplifique o determinante.

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Etapa 1.4.2.1

Simplifique cada termo.

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Etapa 1.4.2.1.1

Multiplique $10$ por $100$ .

$4 \cdot 620 - 10 \cdot 540 + 30 (1000 - 30 \cdot 30)$

Etapa 1.4.2.1.2

Multiplique $- 30$ por $30$ .

$4 \cdot 620 - 10 \cdot 540 + 30 (1000 - 900)$

Etapa 1.4.2.2

Subtraia $900$ de $1000$ .

$4 \cdot 620 - 10 \cdot 540 + 30 \cdot 100$

Etapa 1.5

Simplifique o determinante.

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Etapa 1.5.1

Simplifique cada termo.

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Etapa 1.5.1.1

Multiplique $4$ por $620$ .

$2480 - 10 \cdot 540 + 30 \cdot 100$

Etapa 1.5.1.2

Multiplique $- 10$ por $540$ .

$2480 - 5400 + 30 \cdot 100$

Etapa 1.5.1.3

Multiplique $30$ por $100$ .

$2480 - 5400 + 3000$

Etapa 1.5.2

Subtraia $5400$ de $2480$ .

$- 2920 + 3000$

Etapa 1.5.3

Some $- 2920$ e $3000$ .

$80$

Etapa 2

Since the determinant is non-zero, the inverse exists.

Etapa 3

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} 4 & 10 & 30 & 1 & 0 & 0 \\ 10 & 30 & 100 & 0 & 1 & 0 \\ 30 & 100 & 354 & 0 & 0 & 1 \end{array}]$

Etapa 4

Encontre a forma escalonada reduzida por linhas.

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Etapa 4.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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Etapa 4.1.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{10}{4} & \frac{30}{4} & \frac{1}{4} & \frac{0}{4} & \frac{0}{4} \\ 10 & 30 & 100 & 0 & 1 & 0 \\ 30 & 100 & 354 & 0 & 0 & 1 \end{array}]$

Etapa 4.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 10 & 30 & 100 & 0 & 1 & 0 \\ 30 & 100 & 354 & 0 & 0 & 1 \end{array}]$

Etapa 4.2

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 4.2.1

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 10 - 10 \cdot 1 & 30 - 10 (\frac{5}{2}) & 100 - 10 (\frac{15}{2}) & 0 - 10 (\frac{1}{4}) & 1 - 10 \cdot 0 & 0 - 10 \cdot 0 \\ 30 & 100 & 354 & 0 & 0 & 1 \end{array}]$

Etapa 4.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 5 & 25 & - \frac{5}{2} & 1 & 0 \\ 30 & 100 & 354 & 0 & 0 & 1 \end{array}]$

Etapa 4.3

Perform the row operation $R_{3} = R_{3} - 30 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 4.3.1

Perform the row operation $R_{3} = R_{3} - 30 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 5 & 25 & - \frac{5}{2} & 1 & 0 \\ 30 - 30 \cdot 1 & 100 - 30 (\frac{5}{2}) & 354 - 30 (\frac{15}{2}) & 0 - 30 (\frac{1}{4}) & 0 - 30 \cdot 0 & 1 - 30 \cdot 0 \end{array}]$

Etapa 4.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 5 & 25 & - \frac{5}{2} & 1 & 0 \\ 0 & 25 & 129 & - \frac{15}{2} & 0 & 1 \end{array}]$

Etapa 4.4

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Etapa 4.4.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ \frac{0}{5} & \frac{5}{5} & \frac{25}{5} & \frac{- \frac{5}{2}}{5} & \frac{1}{5} & \frac{0}{5} \\ 0 & 25 & 129 & - \frac{15}{2} & 0 & 1 \end{array}]$

Etapa 4.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 1 & 5 & - \frac{1}{2} & \frac{1}{5} & 0 \\ 0 & 25 & 129 & - \frac{15}{2} & 0 & 1 \end{array}]$

Etapa 4.5

Perform the row operation $R_{3} = R_{3} - 25 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 4.5.1

Perform the row operation $R_{3} = R_{3} - 25 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 1 & 5 & - \frac{1}{2} & \frac{1}{5} & 0 \\ 0 - 25 \cdot 0 & 25 - 25 \cdot 1 & 129 - 25 \cdot 5 & - \frac{15}{2} - 25 (- \frac{1}{2}) & 0 - 25 (\frac{1}{5}) & 1 - 25 \cdot 0 \end{array}]$

Etapa 4.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 1 & 5 & - \frac{1}{2} & \frac{1}{5} & 0 \\ 0 & 0 & 4 & 5 & - 5 & 1 \end{array}]$

Etapa 4.6

Multiply each element of $R_{3}$ by $\frac{1}{4}$ to make the entry at $3, 3$ a $1$ .

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Etapa 4.6.1

Multiply each element of $R_{3}$ by $\frac{1}{4}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 1 & 5 & - \frac{1}{2} & \frac{1}{5} & 0 \\ \frac{0}{4} & \frac{0}{4} & \frac{4}{4} & \frac{5}{4} & \frac{- 5}{4} & \frac{1}{4} \end{array}]$

Etapa 4.6.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 1 & 5 & - \frac{1}{2} & \frac{1}{5} & 0 \\ 0 & 0 & 1 & \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$

Etapa 4.7

Perform the row operation $R_{2} = R_{2} - 5 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 4.7.1

Perform the row operation $R_{2} = R_{2} - 5 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 - 5 \cdot 0 & 1 - 5 \cdot 0 & 5 - 5 \cdot 1 & - \frac{1}{2} - 5 (\frac{5}{4}) & \frac{1}{5} - 5 (- \frac{5}{4}) & 0 - 5 (\frac{1}{4}) \\ 0 & 0 & 1 & \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$

Etapa 4.7.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{15}{2} & \frac{1}{4} & 0 & 0 \\ 0 & 1 & 0 & - \frac{27}{4} & \frac{129}{20} & - \frac{5}{4} \\ 0 & 0 & 1 & \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$

Etapa 4.8

Perform the row operation $R_{1} = R_{1} - \frac{15}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Etapa 4.8.1

Perform the row operation $R_{1} = R_{1} - \frac{15}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{15}{2} \cdot 0 & \frac{5}{2} - \frac{15}{2} \cdot 0 & \frac{15}{2} - \frac{15}{2} \cdot 1 & \frac{1}{4} - \frac{15}{2} \cdot \frac{5}{4} & 0 - \frac{15}{2} (- \frac{5}{4}) & 0 - \frac{15}{2} \cdot \frac{1}{4} \\ 0 & 1 & 0 & - \frac{27}{4} & \frac{129}{20} & - \frac{5}{4} \\ 0 & 0 & 1 & \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$

Etapa 4.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & - \frac{73}{8} & \frac{75}{8} & - \frac{15}{8} \\ 0 & 1 & 0 & - \frac{27}{4} & \frac{129}{20} & - \frac{5}{4} \\ 0 & 0 & 1 & \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$

Etapa 4.9

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 4.9.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{2} \cdot 0 & \frac{5}{2} - \frac{5}{2} \cdot 1 & 0 - \frac{5}{2} \cdot 0 & - \frac{73}{8} - \frac{5}{2} (- \frac{27}{4}) & \frac{75}{8} - \frac{5}{2} \cdot \frac{129}{20} & - \frac{15}{8} - \frac{5}{2} (- \frac{5}{4}) \\ 0 & 1 & 0 & - \frac{27}{4} & \frac{129}{20} & - \frac{5}{4} \\ 0 & 0 & 1 & \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$

Etapa 4.9.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{31}{4} & - \frac{27}{4} & \frac{5}{4} \\ 0 & 1 & 0 & - \frac{27}{4} & \frac{129}{20} & - \frac{5}{4} \\ 0 & 0 & 1 & \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$

Etapa 5

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} \frac{31}{4} & - \frac{27}{4} & \frac{5}{4} \\ - \frac{27}{4} & \frac{129}{20} & - \frac{5}{4} \\ \frac{5}{4} & - \frac{5}{4} & \frac{1}{4} \end{array}]$