Examples

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Step 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{array}{c} \frac{2}{2} & \frac{4}{2} & \frac{6}{2} \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} \frac{2}{2} & \frac{4}{2} & \frac{6}{2} \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

Step 1.1.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 2 & 3 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

[\begin{array}{c} 1 & 2 & 3 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

Step 1.2

Perform the row operation

R_{2} = R_{2} - 8 R_{1}

$R_{2} = R_{2} - 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 1.2.1

Perform the row operation

R_{2} = R_{2} - 8 R_{1}

$R_{2} = R_{2} - 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 2 & 3 \\ 8 - 8 \cdot 1 & 10 - 8 \cdot 2 & 12 - 8 \cdot 3 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 8 - 8 \cdot 1 & 10 - 8 \cdot 2 & 12 - 8 \cdot 3 \\ 1 & 2 & 0 \end{array}]$

Step 1.2.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 & 2 & 0 \end{array}]$

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 & 2 & 0 \end{array}]$

Step 1.3

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Step 1.3.1

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 - 1 & 2 - 2 & 0 - 3 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 - 1 & 2 - 2 & 0 - 3 \end{array}]$

Step 1.3.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 0 & 0 & - 3 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 0 & 0 & - 3 \end{array}]$

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 0 & 0 & - 3 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 0 & 0 & - 3 \end{array}]$

Step 1.4

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{6}

$- \frac{1}{6}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 1.4.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{6}

$- \frac{1}{6}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & 2 & 3 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 12 \\ 0 & 0 & - 3 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 12 \\ 0 & 0 & - 3 \end{array}]$

Step 1.4.2

Simplify

R_{2}

$R_{2}$ .

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

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

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

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

Step 1.5

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Step 1.5.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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

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

Step 1.5.2

Simplify

R_{3}

$R_{3}$ .

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

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

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

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

Step 1.6

Perform the row operation

R_{2} = R_{2} - 2 R_{3}

$R_{2} = R_{2} - 2 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Step 1.6.1

Perform the row operation

R_{2} = R_{2} - 2 R_{3}

$R_{2} = R_{2} - 2 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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

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

Step 1.6.2

Simplify

R_{2}

$R_{2}$ .

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

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

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

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

Step 1.7

Perform the row operation

R_{1} = R_{1} - 3 R_{3}

$R_{1} = R_{1} - 3 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Step 1.7.1

Perform the row operation

R_{1} = R_{1} - 3 R_{3}

$R_{1} = R_{1} - 3 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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

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

Step 1.7.2

Simplify

R_{1}

$R_{1}$ .

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

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

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

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

Step 1.8

Perform the row operation

R_{1} = R_{1} - 2 R_{2}

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Step 1.8.1

Perform the row operation

R_{1} = R_{1} - 2 R_{2}

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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

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

Step 1.8.2

Simplify

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Step 2

The pivot positions are the locations with the leading

1

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

a_{11}, a_{22},

$a_{11}, a_{22},$ and

a_{33}

$a_{33}$

Pivot Columns:

1, 2,

$1, 2,$ and

3

$3$

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