Linear Algebra Examples

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

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify $R_{2}$ .

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

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 $0$ .

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

Step 1.2.2

Simplify $R_{3}$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Simplify $R_{3}$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Simplify $R_{1}$ .

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

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.

$2$