Linear Algebra Examples

$[\begin{array}{c} 3 & - 2 & 0 \\ 8 & - 3 & 4 \\ 13 & - 2 & 0 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{0}{3} \\ 8 & - 3 & 4 \\ 13 & - 2 & 0 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 8 & - 3 & 4 \\ 13 & - 2 & 0 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 8 - 8 \cdot 1 & - 3 - 8 (- \frac{2}{3}) & 4 - 8 \cdot 0 \\ 13 & - 2 & 0 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & \frac{7}{3} & 4 \\ 13 & - 2 & 0 \end{array}]$

Step 3

Perform the row operation $R_{3} = R_{3} - 13 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.1

Perform the row operation $R_{3} = R_{3} - 13 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & \frac{7}{3} & 4 \\ 13 - 13 \cdot 1 & - 2 - 13 (- \frac{2}{3}) & 0 - 13 \cdot 0 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & \frac{7}{3} & 4 \\ 0 & \frac{20}{3} & 0 \end{array}]$

Step 4

Multiply each element of $R_{2}$ by $\frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

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Step 4.1

Multiply each element of $R_{2}$ by $\frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ \frac{3}{7} \cdot 0 & \frac{3}{7} \cdot \frac{7}{3} & \frac{3}{7} \cdot 4 \\ 0 & \frac{20}{3} & 0 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & \frac{12}{7} \\ 0 & \frac{20}{3} & 0 \end{array}]$

Step 5

Perform the row operation $R_{3} = R_{3} - \frac{20}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.1

Perform the row operation $R_{3} = R_{3} - \frac{20}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & \frac{12}{7} \\ 0 - \frac{20}{3} \cdot 0 & \frac{20}{3} - \frac{20}{3} \cdot 1 & 0 - \frac{20}{3} \cdot \frac{12}{7} \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & \frac{12}{7} \\ 0 & 0 & - \frac{80}{7} \end{array}]$

Step 6

Multiply each element of $R_{3}$ by $- \frac{7}{80}$ to make the entry at $3, 3$ a $1$ .

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Step 6.1

Multiply each element of $R_{3}$ by $- \frac{7}{80}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & \frac{12}{7} \\ - \frac{7}{80} \cdot 0 & - \frac{7}{80} \cdot 0 & - \frac{7}{80} (- \frac{80}{7}) \end{array}]$

Step 6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 & 1 & \frac{12}{7} \\ 0 & 0 & 1 \end{array}]$

Step 7

Perform the row operation $R_{2} = R_{2} - \frac{12}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 7.1

Perform the row operation $R_{2} = R_{2} - \frac{12}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 \\ 0 - \frac{12}{7} \cdot 0 & 1 - \frac{12}{7} \cdot 0 & \frac{12}{7} - \frac{12}{7} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 7.2

Simplify $R_{2}$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify $R_{1}$ .

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