Linear Algebra Examples

$[\begin{array}{c} 4 & 4 & 8 & 0 \\ - 12 & - 12 & - 24 & 0 \\ 0 & - 3 & - 9 & 0 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} \frac{4}{4} & \frac{4}{4} & \frac{8}{4} & \frac{0}{4} \\ - 12 & - 12 & - 24 & 0 \\ 0 & - 3 & - 9 & 0 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ - 12 & - 12 & - 24 & 0 \\ 0 & - 3 & - 9 & 0 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} + 12 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} + 12 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ - 12 + 12 \cdot 1 & - 12 + 12 \cdot 1 & - 24 + 12 \cdot 2 & 0 + 12 \cdot 0 \\ 0 & - 3 & - 9 & 0 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - 3 & - 9 & 0 \end{array}]$

Step 3

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ 0 & - 3 & - 9 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4

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

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Step 4.1

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

$[\begin{array}{c} 1 & 1 & 2 & 0 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 9 & - \frac{1}{3} \cdot 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify $R_{1}$ .

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