Linear Algebra Examples

$[\begin{array}{c} 1 & - 4 & 2 \\ - 2 & 8 & - 9 \\ - 1 & 7 & 0 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

$[\begin{array}{c} 1 & - 4 & 2 \\ - 2 + 2 \cdot 1 & 8 + 2 \cdot - 4 & - 9 + 2 \cdot 2 \\ - 1 & 7 & 0 \end{array}]$

Step 1.1.2

Simplify $R_{2}$ .

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

Step 1.2

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

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Step 1.2.1

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

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 0 & - 5 \\ - 1 + 1 \cdot 1 & 7 - 4 & 0 + 1 \cdot 2 \end{array}]$

Step 1.2.2

Simplify $R_{3}$ .

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

Step 1.3

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Simplify $R_{2}$ .

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

Step 1.5

Multiply each element of $R_{3}$ by $- \frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

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Step 1.5.1

Multiply each element of $R_{3}$ by $- \frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

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

Step 1.5.2

Simplify $R_{3}$ .

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

Step 1.6

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

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Step 1.6.1

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

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

Step 1.6.2

Simplify $R_{2}$ .

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

Simplify $R_{1}$ .

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

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

Simplify $R_{1}$ .

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

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: $a_{11}, a_{22},$ and $a_{33}$

Pivot Columns: $1, 2,$ and $3$

Step 3

The rank is the number of pivot columns.

$3$