Linear Algebra Examples

$[\begin{array}{c} 1 & 0 & - 5 & 0 & - 14 \\ 0 & 1 & 2 & 0 & - 7 \\ 0 & 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 & 8 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

Multiply each element of $R_{4}$ by $\frac{1}{8}$ to make the entry at $4, 5$ a $1$ .

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Step 1.1.1

Multiply each element of $R_{4}$ by $\frac{1}{8}$ to make the entry at $4, 5$ a $1$ .

$[\begin{array}{c} 1 & 0 & - 5 & 0 & - 14 \\ 0 & 1 & 2 & 0 & - 7 \\ 0 & 0 & 0 & 1 & 7 \\ \frac{0}{8} & \frac{0}{8} & \frac{0}{8} & \frac{0}{8} & \frac{8}{8} \end{array}]$

Step 1.1.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & - 5 & 0 & - 14 \\ 0 & 1 & 2 & 0 & - 7 \\ 0 & 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 1.2

Perform the row operation $R_{3} = R_{3} - 7 R_{4}$ to make the entry at $3, 5$ a $0$ .

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Step 1.2.1

Perform the row operation $R_{3} = R_{3} - 7 R_{4}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 5 & 0 & - 14 \\ 0 & 1 & 2 & 0 & - 7 \\ 0 - 7 \cdot 0 & 0 - 7 \cdot 0 & 0 - 7 \cdot 0 & 1 - 7 \cdot 0 & 7 - 7 \cdot 1 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 1.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & - 5 & 0 & - 14 \\ 0 & 1 & 2 & 0 & - 7 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 1.3

Perform the row operation $R_{2} = R_{2} + 7 R_{4}$ to make the entry at $2, 5$ a $0$ .

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Step 1.3.1

Perform the row operation $R_{2} = R_{2} + 7 R_{4}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & 0 & - 5 & 0 & - 14 \\ 0 + 7 \cdot 0 & 1 + 7 \cdot 0 & 2 + 7 \cdot 0 & 0 + 7 \cdot 0 & - 7 + 7 \cdot 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 1.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 5 & 0 & - 14 \\ 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 1.4

Perform the row operation $R_{1} = R_{1} + 14 R_{4}$ to make the entry at $1, 5$ a $0$ .

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Step 1.4.1

Perform the row operation $R_{1} = R_{1} + 14 R_{4}$ to make the entry at $1, 5$ a $0$ .

$[\begin{array}{c} 1 + 14 \cdot 0 & 0 + 14 \cdot 0 & - 5 + 14 \cdot 0 & 0 + 14 \cdot 0 & - 14 + 14 \cdot 1 \\ 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 1.4.2

Simplify $R_{1}$ .

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

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: $a_{11}, a_{22}, a_{34},$ and $a_{45}$

Pivot Columns: $1, 2, 4,$ and $5$