Algebra Examples

$[\begin{array}{c} 6 & - 7 \\ 8 & - 9 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

$[\begin{array}{c} \frac{6}{6} & \frac{- 7}{6} \\ 8 & - 9 \end{array}]$

Step 1.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{7}{6} \\ 8 & - 9 \end{array}]$

Step 1.2

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 1.2.1

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

$[\begin{array}{c} 1 & - \frac{7}{6} \\ 8 - 8 \cdot 1 & - 9 - 8 (- \frac{7}{6}) \end{array}]$

Step 1.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{7}{6} \\ 0 & \frac{1}{3} \end{array}]$

Step 1.3

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

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Step 1.3.1

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

$[\begin{array}{c} 1 & - \frac{7}{6} \\ 3 \cdot 0 & 3 (\frac{1}{3}) \end{array}]$

Step 1.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{7}{6} \\ 0 & 1 \end{array}]$

Step 1.4

Perform the row operation $R_{1} = R_{1} + \frac{7}{6} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 1.4.1

Perform the row operation $R_{1} = R_{1} + \frac{7}{6} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{7}{6} \cdot 0 & - \frac{7}{6} + \frac{7}{6} \cdot 1 \\ 0 & 1 \end{array}]$

Step 1.4.2

Simplify $R_{1}$ .

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

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: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$

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