Matemática discreta Ejemplos

$[\begin{array}{c} 0.9 & 0.1 & 0 \\ 0.15 & 0.6 & 0.25 \\ 0 & 0.15 & 0.85 \end{array}]$

Paso 1

Find the determinant.

Toca para ver más pasos...

Paso 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.

Toca para ver más pasos...

Paso 1.1.1

Consider the corresponding sign chart.

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

Paso 1.1.2

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

Paso 1.1.3

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

$| \begin{array}{c} 0.6 & 0.25 \\ 0.15 & 0.85 \end{array} |$

Paso 1.1.4

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

$0.9 | \begin{array}{c} 0.6 & 0.25 \\ 0.15 & 0.85 \end{array} |$

Paso 1.1.5

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

$| \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} |$

Paso 1.1.6

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

$- 0.1 | \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} |$

Paso 1.1.7

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

$| \begin{array}{c} 0.15 & 0.6 \\ 0 & 0.15 \end{array} |$

Paso 1.1.8

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

$0 | \begin{array}{c} 0.15 & 0.6 \\ 0 & 0.15 \end{array} |$

Paso 1.1.9

Add the terms together.

$0.9 | \begin{array}{c} 0.6 & 0.25 \\ 0.15 & 0.85 \end{array} | - 0.1 | \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} | + 0 | \begin{array}{c} 0.15 & 0.6 \\ 0 & 0.15 \end{array} |$

Paso 1.2

Multiplica $0$ por $| \begin{array}{c} 0.15 & 0.6 \\ 0 & 0.15 \end{array} |$ .

$0.9 | \begin{array}{c} 0.6 & 0.25 \\ 0.15 & 0.85 \end{array} | - 0.1 | \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} | + 0$

Paso 1.3

Evalúa $| \begin{array}{c} 0.6 & 0.25 \\ 0.15 & 0.85 \end{array} |$ .

Toca para ver más pasos...

Paso 1.3.1

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

$0.9 (0.6 \cdot 0.85 - 0.15 \cdot 0.25) - 0.1 | \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} | + 0$

Paso 1.3.2

Simplifica el determinante.

Toca para ver más pasos...

Paso 1.3.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 1.3.2.1.1

Multiplica $0.6$ por $0.85$ .

$0.9 (0.51 - 0.15 \cdot 0.25) - 0.1 | \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} | + 0$

Paso 1.3.2.1.2

Multiplica $- 0.15$ por $0.25$ .

$0.9 (0.51 - 0.0375) - 0.1 | \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} | + 0$

Paso 1.3.2.2

Resta $0.0375$ de $0.51$ .

$0.9 \cdot 0.4725 - 0.1 | \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} | + 0$

Paso 1.4

Evalúa $| \begin{array}{c} 0.15 & 0.25 \\ 0 & 0.85 \end{array} |$ .

Toca para ver más pasos...

Paso 1.4.1

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

$0.9 \cdot 0.4725 - 0.1 (0.15 \cdot 0.85 + 0 \cdot 0.25) + 0$

Paso 1.4.2

Simplifica el determinante.

Toca para ver más pasos...

Paso 1.4.2.1

Simplifica cada término.

Toca para ver más pasos...

Paso 1.4.2.1.1

Multiplica $0.15$ por $0.85$ .

$0.9 \cdot 0.4725 - 0.1 (0.1275 + 0 \cdot 0.25) + 0$

Paso 1.4.2.1.2

Multiplica $0$ por $0.25$ .

$0.9 \cdot 0.4725 - 0.1 (0.1275 + 0) + 0$

Paso 1.4.2.2

Suma $0.1275$ y $0$ .

$0.9 \cdot 0.4725 - 0.1 \cdot 0.1275 + 0$

Paso 1.5

Simplifica el determinante.

Toca para ver más pasos...

Paso 1.5.1

Simplifica cada término.

Toca para ver más pasos...

Paso 1.5.1.1

Multiplica $0.9$ por $0.4725$ .

$0.42525 - 0.1 \cdot 0.1275 + 0$

Paso 1.5.1.2

Multiplica $- 0.1$ por $0.1275$ .

$0.42525 - 0.01275 + 0$

Paso 1.5.2

Resta $0.01275$ de $0.42525$ .

$0.4125 + 0$

Paso 1.5.3

Suma $0.4125$ y $0$ .

$0.4125$

Paso 2

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

Paso 3

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

$[\begin{array}{c} 0.9 & 0.1 & 0 & 1 & 0 & 0 \\ 0.15 & 0.6 & 0.25 & 0 & 1 & 0 \\ 0 & 0.15 & 0.85 & 0 & 0 & 1 \end{array}]$

Paso 4

Obtén la forma escalonada reducida por filas.

Toca para ver más pasos...

Paso 4.1

Multiply each element of $R_{1}$ by $\frac{1}{0.9}$ to make the entry at $1, 1$ a $1$ .

Toca para ver más pasos...

Paso 4.1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.9}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.9}{0.9} & \frac{0.1}{0.9} & \frac{0}{0.9} & \frac{1}{0.9} & \frac{0}{0.9} & \frac{0}{0.9} \\ 0.15 & 0.6 & 0.25 & 0 & 1 & 0 \\ 0 & 0.15 & 0.85 & 0 & 0 & 1 \end{array}]$

Paso 4.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0.15 & 0.6 & 0.25 & 0 & 1 & 0 \\ 0 & 0.15 & 0.85 & 0 & 0 & 1 \end{array}]$

Paso 4.2

Perform the row operation $R_{2} = R_{2} - 0.15 R_{1}$ to make the entry at $2, 1$ a $0$ .

Toca para ver más pasos...

Paso 4.2.1

Perform the row operation $R_{2} = R_{2} - 0.15 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0.15 - 0.15 \cdot 1 & 0.6 - 0.15 \cdot 0.\overline{1} & 0.25 - 0.15 \cdot 0 & 0 - 0.15 \cdot 1.\overline{1} & 1 - 0.15 \cdot 0 & 0 - 0.15 \cdot 0 \\ 0 & 0.15 & 0.85 & 0 & 0 & 1 \end{array}]$

Paso 4.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 & 0.58\overline{3} & 0.25 & - 0.1\overline{6} & 1 & 0 \\ 0 & 0.15 & 0.85 & 0 & 0 & 1 \end{array}]$

Paso 4.3

Multiply each element of $R_{2}$ by $\frac{1}{0.58\overline{3}}$ to make the entry at $2, 2$ a $1$ .

Toca para ver más pasos...

Paso 4.3.1

Multiply each element of $R_{2}$ by $\frac{1}{0.58\overline{3}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ \frac{0}{0.58\overline{3}} & \frac{0.58\overline{3}}{0.58\overline{3}} & \frac{0.25}{0.58\overline{3}} & \frac{- 0.1\overline{6}}{0.58\overline{3}} & \frac{1}{0.58\overline{3}} & \frac{0}{0.58\overline{3}} \\ 0 & 0.15 & 0.85 & 0 & 0 & 1 \end{array}]$

Paso 4.3.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 & 1 & 0.\overline{428571} & - 0.\overline{285714} & 1.\overline{714285} & 0 \\ 0 & 0.15 & 0.85 & 0 & 0 & 1 \end{array}]$

Paso 4.4

Perform the row operation $R_{3} = R_{3} - 0.15 R_{2}$ to make the entry at $3, 2$ a $0$ .

Toca para ver más pasos...

Paso 4.4.1

Perform the row operation $R_{3} = R_{3} - 0.15 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 & 1 & 0.\overline{428571} & - 0.\overline{285714} & 1.\overline{714285} & 0 \\ 0 - 0.15 \cdot 0 & 0.15 - 0.15 \cdot 1 & 0.85 - 0.15 \cdot 0.\overline{428571} & 0 - 0.15 \cdot - 0.\overline{285714} & 0 - 0.15 \cdot 1.\overline{714285} & 1 - 0.15 \cdot 0 \end{array}]$

Paso 4.4.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 & 1 & 0.\overline{428571} & - 0.\overline{285714} & 1.\overline{714285} & 0 \\ 0 & 0 & 0.7\overline{857142} & 0.0\overline{428571} & - 0.2\overline{571428} & 1 \end{array}]$

Paso 4.5

Multiply each element of $R_{3}$ by $\frac{1}{0.7\overline{857142}}$ to make the entry at $3, 3$ a $1$ .

Toca para ver más pasos...

Paso 4.5.1

Multiply each element of $R_{3}$ by $\frac{1}{0.7\overline{857142}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 & 1 & 0.\overline{428571} & - 0.\overline{285714} & 1.\overline{714285} & 0 \\ \frac{0}{0.7\overline{857142}} & \frac{0}{0.7\overline{857142}} & \frac{0.7\overline{857142}}{0.7\overline{857142}} & \frac{0.0\overline{428571}}{0.7\overline{857142}} & \frac{- 0.2\overline{571428}}{0.7\overline{857142}} & \frac{1}{0.7\overline{857142}} \end{array}]$

Paso 4.5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 & 1 & 0.\overline{428571} & - 0.\overline{285714} & 1.\overline{714285} & 0 \\ 0 & 0 & 1 & 0.0\overline{54} & - 0.3\overline{27} & 1.\overline{27} \end{array}]$

Paso 4.6

Perform the row operation $R_{2} = R_{2} - 0.\overline{428571} R_{3}$ to make the entry at $2, 3$ a $0$ .

Toca para ver más pasos...

Paso 4.6.1

Perform the row operation $R_{2} = R_{2} - 0.\overline{428571} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 - 0.\overline{428571} \cdot 0 & 1 - 0.\overline{428571} \cdot 0 & 0.\overline{428571} - 0.\overline{428571} \cdot 1 & - 0.\overline{285714} - 0.\overline{428571} \cdot 0.0\overline{54} & 1.\overline{714285} - 0.\overline{428571} \cdot - 0.3\overline{27} & 0 - 0.\overline{428571} \cdot 1.\overline{27} \\ 0 & 0 & 1 & 0.0\overline{54} & - 0.3\overline{27} & 1.\overline{27} \end{array}]$

Paso 4.6.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0.\overline{1} & 0 & 1.\overline{1} & 0 & 0 \\ 0 & 1 & 0 & - 0.3\overline{09} & 1.8\overline{54} & - 0.\overline{54} \\ 0 & 0 & 1 & 0.0\overline{54} & - 0.3\overline{27} & 1.\overline{27} \end{array}]$

Paso 4.7

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

Toca para ver más pasos...

Paso 4.7.1

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

$[\begin{array}{c} 1 - 0.\overline{1} \cdot 0 & 0.\overline{1} - 0.\overline{1} \cdot 1 & 0 - 0.\overline{1} \cdot 0 & 1.\overline{1} - 0.\overline{1} \cdot - 0.3\overline{09} & 0 - 0.\overline{1} \cdot 1.8\overline{54} & 0 - 0.\overline{1} \cdot - 0.\overline{54} \\ 0 & 1 & 0 & - 0.3\overline{09} & 1.8\overline{54} & - 0.\overline{54} \\ 0 & 0 & 1 & 0.0\overline{54} & - 0.3\overline{27} & 1.\overline{27} \end{array}]$

Paso 4.7.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1.1\overline{45} & - 0.2\overline{06} & 0.\overline{06} \\ 0 & 1 & 0 & - 0.3\overline{09} & 1.8\overline{54} & - 0.\overline{54} \\ 0 & 0 & 1 & 0.0\overline{54} & - 0.3\overline{27} & 1.\overline{27} \end{array}]$

Paso 5

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

$[\begin{array}{c} 1.1\overline{45} & - 0.2\overline{06} & 0.\overline{06} \\ - 0.3\overline{09} & 1.8\overline{54} & - 0.\overline{54} \\ 0.0\overline{54} & - 0.3\overline{27} & 1.\overline{27} \end{array}]$