Algebra lineare Esempi

$[\begin{array}{c} a + b & 2 a & - a \\ c + d & 2 c & - c \end{array}]$

Passaggio 1

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 1.1

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

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Passaggio 1.1.1

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

$[\begin{array}{c} \frac{a + b}{a + b} & \frac{2 a}{a + b} & \frac{- a}{a + b} \\ c + d & 2 c & - c \end{array}]$

Passaggio 1.1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2 a}{a + b} & - \frac{a}{a + b} \\ c + d & 2 c & - c \end{array}]$

Passaggio 1.2

Perform the row operation $R_{2} = R_{2} - (c + d) R_{1}$ to make the entry at $2, 1$ a $0$ .

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Passaggio 1.2.1

Perform the row operation $R_{2} = R_{2} - (c + d) R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2 a}{a + b} & - \frac{a}{a + b} \\ c + d - (c + d) \cdot 1 & 2 c - (c + d) \frac{2 a}{a + b} & - c - (c + d) (- \frac{a}{a + b}) \end{array}]$

Passaggio 1.2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2 a}{a + b} & - \frac{a}{a + b} \\ 0 & \frac{2 (c b - d a)}{a + b} & - \frac{c b - a d}{a + b} \end{array}]$

Passaggio 1.3

Multiply each element of $R_{2}$ by $\frac{a + b}{2 (c b - d a)}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 1.3.1

Multiply each element of $R_{2}$ by $\frac{a + b}{2 (c b - d a)}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2 a}{a + b} & - \frac{a}{a + b} \\ \frac{a + b}{2 (c b - d a)} \cdot 0 & \frac{a + b}{2 (c b - d a)} \cdot \frac{2 (c b - d a)}{a + b} & \frac{a + b}{2 (c b - d a)} (- \frac{c b - a d}{a + b}) \end{array}]$

Passaggio 1.3.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2 a}{a + b} & - \frac{a}{a + b} \\ 0 & 1 & - \frac{b c}{2 (c b - d a)} + \frac{a d}{2 (c b - d a)} \end{array}]$

Passaggio 1.4

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

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Passaggio 1.4.1

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

$[\begin{array}{c} 1 - \frac{2 a}{a + b} \cdot 0 & \frac{2 a}{a + b} - \frac{2 a}{a + b} \cdot 1 & - \frac{a}{a + b} - \frac{2 a}{a + b} (- \frac{b c}{2 (c b - d a)} + \frac{a d}{2 (c b - d a)}) \\ 0 & 1 & - \frac{b c}{2 (c b - d a)} + \frac{a d}{2 (c b - d a)} \end{array}]$

Passaggio 1.4.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & - \frac{b c}{2 (c b - d a)} + \frac{a d}{2 (c b - d a)} \end{array}]$

Passaggio 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$