Algebra lineare Esempi

$A = [\begin{array}{c} i & - 5 & 0 \\ 1 & i & - 1 \end{array}]$

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 $\frac{1}{i}$ to make the entry at $1, 1$ a $1$ .

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Passaggio 1.1.1

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

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

Passaggio 1.1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 5 i & 0 \\ 1 & i & - 1 \end{array}]$

Passaggio 1.2

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

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Passaggio 1.2.1

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

$[\begin{array}{c} 1 & 5 i & 0 \\ 1 - 1 & i - 5 i & - 1 - 0 \end{array}]$

Passaggio 1.2.2

Semplifica $R_{2}$ .

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

Passaggio 1.3

Multiply each element of $R_{2}$ by $\frac{1}{- 4 i}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 1.3.1

Multiply each element of $R_{2}$ by $\frac{1}{- 4 i}$ to make the entry at $2, 2$ a $1$ .

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

Passaggio 1.3.2

Semplifica $R_{2}$ .

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

Passaggio 1.4

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

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Passaggio 1.4.1

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

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

Passaggio 1.4.2

Semplifica $R_{1}$ .

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

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 $2$

Passaggio 3

The rank is the number of pivot columns.

$2$