Linear Algebra Examples

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

Step 1

Find the reduced row echelon form.

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Step 1.1

Perform the row operation $R_{1} = R_{1} + 2 R_{3}$ to make the entry at $1, 5$ a $0$ .

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Step 1.1.1

Perform the row operation $R_{1} = R_{1} + 2 R_{3}$ to make the entry at $1, 5$ a $0$ .

$[\begin{array}{c} 1 + 2 \cdot 0 & 0 + 2 \cdot 0 & - 4 + 2 \cdot 0 & 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 3 + 2 \cdot 0 \\ 0 & 1 & 7 & - 1 & 0 & 5 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 1.1.2

Simplify $R_{1}$ .

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

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: $a_{11}, a_{22},$ and $a_{35}$

Pivot Columns: $1, 2,$ and $5$

Step 3

The rank is the number of pivot columns.

$3$