Linear Algebra Examples

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 3 & - 8 & h \\ - 3 & 6 & - 7 & k \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 3 & - 8 & h \\ - 3 + 3 \cdot 1 & 6 + 3 \cdot - 3 & - 7 + 3 \cdot 5 & k + 3 g \end{array}]$

Step 1.1.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 3 & - 8 & h \\ 0 & - 3 & 8 & k + 3 g \end{array}]$

Step 1.2

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

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Step 1.2.1

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

$[\begin{array}{c} 1 & - 3 & 5 & g \\ \frac{0}{3} & \frac{3}{3} & \frac{- 8}{3} & \frac{h}{3} \\ 0 & - 3 & 8 & k + 3 g \end{array}]$

Step 1.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 1 & - \frac{8}{3} & \frac{h}{3} \\ 0 & - 3 & 8 & k + 3 g \end{array}]$

Step 1.3

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

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Step 1.3.1

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

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 1 & - \frac{8}{3} & \frac{h}{3} \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 8 + 3 (- \frac{8}{3}) & k + 3 g + 3 \frac{h}{3} \end{array}]$

Step 1.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 1 & - \frac{8}{3} & \frac{h}{3} \\ 0 & 0 & 0 & k + 3 g + h \end{array}]$

Step 1.4

Multiply each element of $R_{3}$ by $\frac{1}{k + 3 g + h}$ to make the entry at $3, 4$ a $1$ .

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Step 1.4.1

Multiply each element of $R_{3}$ by $\frac{1}{k + 3 g + h}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 1 & - \frac{8}{3} & \frac{h}{3} \\ \frac{0}{k + 3 g + h} & \frac{0}{k + 3 g + h} & \frac{0}{k + 3 g + h} & \frac{k + 3 g + h}{k + 3 g + h} \end{array}]$

Step 1.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 1 & - \frac{8}{3} & \frac{h}{3} \\ 0 & 0 & 0 & 1 \end{array}]$

Step 1.5

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

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Step 1.5.1

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

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 - \frac{h}{3} \cdot 0 & 1 - \frac{h}{3} \cdot 0 & - \frac{8}{3} - \frac{h}{3} \cdot 0 & \frac{h}{3} - \frac{h}{3} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 1.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 5 & g \\ 0 & 1 & - \frac{8}{3} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 1.6

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

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Step 1.6.1

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

$[\begin{array}{c} 1 - g \cdot 0 & - 3 - g \cdot 0 & 5 - g \cdot 0 & g - g \cdot 1 \\ 0 & 1 & - \frac{8}{3} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 1.6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 3 & 5 & 0 \\ 0 & 1 & - \frac{8}{3} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 1.7

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 1.7.1

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

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

Step 1.7.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 3 & 0 \\ 0 & 1 & - \frac{8}{3} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

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

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