Finite Math Examples

$[\begin{array}{c} 2 & 1 & 4 & 1 & 0 & 0 & 32 \\ 2 & 4 & 1 & 0 & 1 & 0 & 48 \\ - 4 & - 3 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{1}{2} & \frac{4}{2} & \frac{1}{2} & \frac{0}{2} & \frac{0}{2} & \frac{32}{2} \\ 2 & 4 & 1 & 0 & 1 & 0 & 48 \\ - 4 & - 3 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 2 & 4 & 1 & 0 & 1 & 0 & 48 \\ - 4 & - 3 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.2

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

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Step 1.2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 2 - 2 \cdot 1 & 4 - 2 (\frac{1}{2}) & 1 - 2 \cdot 2 & 0 - 2 (\frac{1}{2}) & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 & 48 - 2 \cdot 16 \\ - 4 & - 3 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 3 & - 3 & - 1 & 1 & 0 & 16 \\ - 4 & - 3 & - 2 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.3

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

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Step 1.3.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 3 & - 3 & - 1 & 1 & 0 & 16 \\ - 4 + 4 \cdot 1 & - 3 + 4 (\frac{1}{2}) & - 2 + 4 \cdot 2 & 0 + 4 (\frac{1}{2}) & 0 + 4 \cdot 0 & 1 + 4 \cdot 0 & 0 + 4 \cdot 16 \end{array}]$

Step 1.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 3 & - 3 & - 1 & 1 & 0 & 16 \\ 0 & - 1 & 6 & 2 & 0 & 1 & 64 \end{array}]$

Step 1.4

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

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Step 1.4.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ \frac{0}{3} & \frac{3}{3} & \frac{- 3}{3} & \frac{- 1}{3} & \frac{1}{3} & \frac{0}{3} & \frac{16}{3} \\ 0 & - 1 & 6 & 2 & 0 & 1 & 64 \end{array}]$

Step 1.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 1 & - 1 & - \frac{1}{3} & \frac{1}{3} & 0 & \frac{16}{3} \\ 0 & - 1 & 6 & 2 & 0 & 1 & 64 \end{array}]$

Step 1.5

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

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Step 1.5.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 1 & - 1 & - \frac{1}{3} & \frac{1}{3} & 0 & \frac{16}{3} \\ 0 + 0 & - 1 + 1 \cdot 1 & 6 - 1 & 2 - \frac{1}{3} & 0 + \frac{1}{3} & 1 + 0 & 64 + \frac{16}{3} \end{array}]$

Step 1.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 1 & - 1 & - \frac{1}{3} & \frac{1}{3} & 0 & \frac{16}{3} \\ 0 & 0 & 5 & \frac{5}{3} & \frac{1}{3} & 1 & \frac{208}{3} \end{array}]$

Step 1.6

Multiply each element of $R_{3}$ by $\frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

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Step 1.6.1

Multiply each element of $R_{3}$ by $\frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 1 & - 1 & - \frac{1}{3} & \frac{1}{3} & 0 & \frac{16}{3} \\ \frac{0}{5} & \frac{0}{5} & \frac{5}{5} & \frac{\frac{5}{3}}{5} & \frac{\frac{1}{3}}{5} & \frac{1}{5} & \frac{\frac{208}{3}}{5} \end{array}]$

Step 1.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 1 & - 1 & - \frac{1}{3} & \frac{1}{3} & 0 & \frac{16}{3} \\ 0 & 0 & 1 & \frac{1}{3} & \frac{1}{15} & \frac{1}{5} & \frac{208}{15} \end{array}]$

Step 1.7

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

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Step 1.7.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & - \frac{1}{3} + \frac{1}{3} & \frac{1}{3} + \frac{1}{15} & 0 + \frac{1}{5} & \frac{16}{3} + \frac{208}{15} \\ 0 & 0 & 1 & \frac{1}{3} & \frac{1}{15} & \frac{1}{5} & \frac{208}{15} \end{array}]$

Step 1.7.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 2 & \frac{1}{2} & 0 & 0 & 16 \\ 0 & 1 & 0 & 0 & \frac{2}{5} & \frac{1}{5} & \frac{96}{5} \\ 0 & 0 & 1 & \frac{1}{3} & \frac{1}{15} & \frac{1}{5} & \frac{208}{15} \end{array}]$

Step 1.8

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

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Step 1.8.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & \frac{1}{2} - 2 \cdot 0 & 2 - 2 \cdot 1 & \frac{1}{2} - 2 (\frac{1}{3}) & 0 - 2 (\frac{1}{15}) & 0 - 2 (\frac{1}{5}) & 16 - 2 (\frac{208}{15}) \\ 0 & 1 & 0 & 0 & \frac{2}{5} & \frac{1}{5} & \frac{96}{5} \\ 0 & 0 & 1 & \frac{1}{3} & \frac{1}{15} & \frac{1}{5} & \frac{208}{15} \end{array}]$

Step 1.8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{1}{6} & - \frac{2}{15} & - \frac{2}{5} & - \frac{176}{15} \\ 0 & 1 & 0 & 0 & \frac{2}{5} & \frac{1}{5} & \frac{96}{5} \\ 0 & 0 & 1 & \frac{1}{3} & \frac{1}{15} & \frac{1}{5} & \frac{208}{15} \end{array}]$

Step 1.9

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

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Step 1.9.1

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

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 0 - \frac{1}{2} \cdot 0 & - \frac{1}{6} - \frac{1}{2} \cdot 0 & - \frac{2}{15} - \frac{1}{2} \cdot \frac{2}{5} & - \frac{2}{5} - \frac{1}{2} \cdot \frac{1}{5} & - \frac{176}{15} - \frac{1}{2} \cdot \frac{96}{5} \\ 0 & 1 & 0 & 0 & \frac{2}{5} & \frac{1}{5} & \frac{96}{5} \\ 0 & 0 & 1 & \frac{1}{3} & \frac{1}{15} & \frac{1}{5} & \frac{208}{15} \end{array}]$

Step 1.9.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{6} & - \frac{1}{3} & - \frac{1}{2} & - \frac{64}{3} \\ 0 & 1 & 0 & 0 & \frac{2}{5} & \frac{1}{5} & \frac{96}{5} \\ 0 & 0 & 1 & \frac{1}{3} & \frac{1}{15} & \frac{1}{5} & \frac{208}{15} \end{array}]$

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 $a_{33}$

Pivot Columns: $1, 2,$ and $3$