Precalculus Examples

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

Step 1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Step 1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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

Step 1.2

Simplify $R_{1}$ .

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

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 $0$ .

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

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & - 8 & 1 & \frac{20}{3} \\ 3 & 9 & 0 & 2 \end{array}]$

Step 3

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 3.1

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

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & - 8 & 1 & \frac{20}{3} \\ 3 - 3 \cdot 1 & 9 - 3 \cdot 1 & 0 - 3 \cdot 1 & 2 - 3 (- \frac{1}{3}) \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & - 8 & 1 & \frac{20}{3} \\ 0 & 6 & - 3 & 3 \end{array}]$

Step 4

Multiply each element of $R_{2}$ by $- \frac{1}{8}$ to make the entry at $2, 2$ a $1$ .

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Step 4.1

Multiply each element of $R_{2}$ by $- \frac{1}{8}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot - 8 & - \frac{1}{8} \cdot 1 & - \frac{1}{8} \cdot \frac{20}{3} \\ 0 & 6 & - 3 & 3 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & 1 & - \frac{1}{8} & - \frac{5}{6} \\ 0 & 6 & - 3 & 3 \end{array}]$

Step 5

Perform the row operation $R_{3} = R_{3} - 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.1

Perform the row operation $R_{3} = R_{3} - 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & 1 & - \frac{1}{8} & - \frac{5}{6} \\ 0 - 6 \cdot 0 & 6 - 6 \cdot 1 & - 3 - 6 (- \frac{1}{8}) & 3 - 6 (- \frac{5}{6}) \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & 1 & - \frac{1}{8} & - \frac{5}{6} \\ 0 & 0 & - \frac{9}{4} & 8 \end{array}]$

Step 6

Multiply each element of $R_{3}$ by $- \frac{4}{9}$ to make the entry at $3, 3$ a $1$ .

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Step 6.1

Multiply each element of $R_{3}$ by $- \frac{4}{9}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & 1 & - \frac{1}{8} & - \frac{5}{6} \\ - \frac{4}{9} \cdot 0 & - \frac{4}{9} \cdot 0 & - \frac{4}{9} (- \frac{9}{4}) & - \frac{4}{9} \cdot 8 \end{array}]$

Step 6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & 1 & - \frac{1}{8} & - \frac{5}{6} \\ 0 & 0 & 1 & - \frac{32}{9} \end{array}]$

Step 7

Perform the row operation $R_{2} = R_{2} + \frac{1}{8} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 7.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{8} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 + \frac{1}{8} \cdot 0 & 1 + \frac{1}{8} \cdot 0 & - \frac{1}{8} + \frac{1}{8} \cdot 1 & - \frac{5}{6} + \frac{1}{8} (- \frac{32}{9}) \\ 0 & 0 & 1 & - \frac{32}{9} \end{array}]$

Step 7.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{23}{18} \\ 0 & 0 & 1 & - \frac{32}{9} \end{array}]$

Step 8

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

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Step 8.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & - \frac{1}{3} + \frac{32}{9} \\ 0 & 1 & 0 & - \frac{23}{18} \\ 0 & 0 & 1 & - \frac{32}{9} \end{array}]$

Step 8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & \frac{29}{9} \\ 0 & 1 & 0 & - \frac{23}{18} \\ 0 & 0 & 1 & - \frac{32}{9} \end{array}]$

Step 9

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

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Step 9.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & \frac{29}{9} + \frac{23}{18} \\ 0 & 1 & 0 & - \frac{23}{18} \\ 0 & 0 & 1 & - \frac{32}{9} \end{array}]$

Step 9.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{9}{2} \\ 0 & 1 & 0 & - \frac{23}{18} \\ 0 & 0 & 1 & - \frac{32}{9} \end{array}]$