Álgebra lineal Ejemplos

$3 a - 2 b = 1$ , $4 a + 5 b = 3$ , $7 a + 3 b = 2$

Paso 1

Escribe el sistema de ecuaciones en forma de matriz.

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

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{1}{3} \\ 4 & 5 & 3 \\ 7 & 3 & 2 \end{array}]$

Paso 2.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 4 & 5 & 3 \\ 7 & 3 & 2 \end{array}]$

Paso 2.2

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 2.2.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 4 - 4 \cdot 1 & 5 - 4 (- \frac{2}{3}) & 3 - 4 (\frac{1}{3}) \\ 7 & 3 & 2 \end{array}]$

Paso 2.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & \frac{23}{3} & \frac{5}{3} \\ 7 & 3 & 2 \end{array}]$

Paso 2.3

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 2.3.1

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & \frac{23}{3} & \frac{5}{3} \\ 7 - 7 \cdot 1 & 3 - 7 (- \frac{2}{3}) & 2 - 7 (\frac{1}{3}) \end{array}]$

Paso 2.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & \frac{23}{3} & \frac{5}{3} \\ 0 & \frac{23}{3} & - \frac{1}{3} \end{array}]$

Paso 2.4

Multiply each element of $R_{2}$ by $\frac{3}{23}$ to make the entry at $2, 2$ a $1$ .

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Paso 2.4.1

Multiply each element of $R_{2}$ by $\frac{3}{23}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ \frac{3}{23} \cdot 0 & \frac{3}{23} \cdot \frac{23}{3} & \frac{3}{23} \cdot \frac{5}{3} \\ 0 & \frac{23}{3} & - \frac{1}{3} \end{array}]$

Paso 2.4.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 1 & \frac{5}{23} \\ 0 & \frac{23}{3} & - \frac{1}{3} \end{array}]$

Paso 2.5

Perform the row operation $R_{3} = R_{3} - \frac{23}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 2.5.1

Perform the row operation $R_{3} = R_{3} - \frac{23}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 1 & \frac{5}{23} \\ 0 - \frac{23}{3} \cdot 0 & \frac{23}{3} - \frac{23}{3} \cdot 1 & - \frac{1}{3} - \frac{23}{3} \cdot \frac{5}{23} \end{array}]$

Paso 2.5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 1 & \frac{5}{23} \\ 0 & 0 & - 2 \end{array}]$

Paso 2.6

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

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Paso 2.6.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 1 & \frac{5}{23} \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 \end{array}]$

Paso 2.6.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 1 & \frac{5}{23} \\ 0 & 0 & 1 \end{array}]$

Paso 2.7

Perform the row operation $R_{2} = R_{2} - \frac{5}{23} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Paso 2.7.1

Perform the row operation $R_{2} = R_{2} - \frac{5}{23} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 - \frac{5}{23} \cdot 0 & 1 - \frac{5}{23} \cdot 0 & \frac{5}{23} - \frac{5}{23} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Paso 2.7.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Paso 2.8

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

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Paso 2.8.1

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

$[\begin{array}{c} 1 - \frac{1}{3} \cdot 0 & - \frac{2}{3} - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Paso 2.8.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Paso 2.9

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

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Paso 2.9.1

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

$[\begin{array}{c} 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 0 + \frac{2}{3} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Paso 2.9.2

Simplifica $R_{1}$ .

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

Paso 3

Usa la matriz de resultados para declarar las soluciones finales en el sistema de ecuaciones.

$a = 0$

$b = 0$

$0 = 1$

Paso 4

Como $0 \neq 1$ , no hay soluciones.

No hay solución