Linear Algebra Examples

$[\begin{array}{c} 0 & 1 & 2 \\ 3 & 4 & 5 \\ 6 & 7 & 8 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 3 & 4 & 5 \\ 0 & 1 & 2 \\ 6 & 7 & 8 \end{array}]$

Step 1.2

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

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Step 1.2.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{4}{3} & \frac{5}{3} \\ 0 & 1 & 2 \\ 6 & 7 & 8 \end{array}]$

Step 1.2.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{5}{3} \\ 0 & 1 & 2 \\ 6 & 7 & 8 \end{array}]$

Step 1.3

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 1.3.1

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{5}{3} \\ 0 & 1 & 2 \\ 6 - 6 \cdot 1 & 7 - 6 (\frac{4}{3}) & 8 - 6 (\frac{5}{3}) \end{array}]$

Step 1.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{5}{3} \\ 0 & 1 & 2 \\ 0 & - 1 & - 2 \end{array}]$

Step 1.4

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

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Step 1.4.1

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

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{5}{3} \\ 0 & 1 & 2 \\ 0 + 0 & - 1 + 1 \cdot 1 & - 2 + 1 \cdot 2 \end{array}]$

Step 1.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{4}{3} & \frac{5}{3} \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array}]$

Step 1.5

Perform the row operation $R_{1} = R_{1} - \frac{4}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 1.5.1

Perform the row operation $R_{1} = R_{1} - \frac{4}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{4}{3} \cdot 0 & \frac{4}{3} - \frac{4}{3} \cdot 1 & \frac{5}{3} - \frac{4}{3} \cdot 2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array}]$

Step 1.5.2

Simplify $R_{1}$ .

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

Step 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 $2$

Step 3

The rank is the number of pivot columns.

$2$