Linear Algebra Examples

$[\begin{array}{c} 3 & - 2 & 4 & 0 \\ 9 & - 6 & 12 & 0 \\ 6 & - 4 & 8 & 0 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{4}{3} & \frac{0}{3} \\ 9 & - 6 & 12 & 0 \\ 6 & - 4 & 8 & 0 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{3} & 0 \\ 9 & - 6 & 12 & 0 \\ 6 & - 4 & 8 & 0 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - 9 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 9 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{3} & 0 \\ 9 - 9 \cdot 1 & - 6 - 9 (- \frac{2}{3}) & 12 - 9 (\frac{4}{3}) & 0 - 9 \cdot 0 \\ 6 & - 4 & 8 & 0 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{3} & 0 \\ 0 & 0 & 0 & 0 \\ 6 & - 4 & 8 & 0 \end{array}]$

Step 3

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.1

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{4}{3} & 0 \\ 0 & 0 & 0 & 0 \\ 6 - 6 \cdot 1 & - 4 - 6 (- \frac{2}{3}) & 8 - 6 (\frac{4}{3}) & 0 - 6 \cdot 0 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

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