Linear Algebra Examples

$[\begin{array}{c} 5 & 9 & 3 \\ 3 & 5 & 6 \\ 1 & 5 & 3 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify $R_{1}$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & - \frac{2}{5} & \frac{21}{5} \\ 1 & 5 & 3 \end{array}]$

Step 1.3

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

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Step 1.3.1

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

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & - \frac{2}{5} & \frac{21}{5} \\ 1 - 1 & 5 - \frac{9}{5} & 3 - \frac{3}{5} \end{array}]$

Step 1.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & - \frac{2}{5} & \frac{21}{5} \\ 0 & \frac{16}{5} & \frac{12}{5} \end{array}]$

Step 1.4

Multiply each element of $R_{2}$ by $- \frac{5}{2}$ to make the entry at $2, 2$ a $1$ .

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Step 1.4.1

Multiply each element of $R_{2}$ by $- \frac{5}{2}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ - \frac{5}{2} \cdot 0 & - \frac{5}{2} (- \frac{2}{5}) & - \frac{5}{2} \cdot \frac{21}{5} \\ 0 & \frac{16}{5} & \frac{12}{5} \end{array}]$

Step 1.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & 1 & - \frac{21}{2} \\ 0 & \frac{16}{5} & \frac{12}{5} \end{array}]$

Step 1.5

Perform the row operation $R_{3} = R_{3} - \frac{16}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 1.5.1

Perform the row operation $R_{3} = R_{3} - \frac{16}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & 1 & - \frac{21}{2} \\ 0 - \frac{16}{5} \cdot 0 & \frac{16}{5} - \frac{16}{5} \cdot 1 & \frac{12}{5} - \frac{16}{5} (- \frac{21}{2}) \end{array}]$

Step 1.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & 1 & - \frac{21}{2} \\ 0 & 0 & 36 \end{array}]$

Step 1.6

Multiply each element of $R_{3}$ by $\frac{1}{36}$ to make the entry at $3, 3$ a $1$ .

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Step 1.6.1

Multiply each element of $R_{3}$ by $\frac{1}{36}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & 1 & - \frac{21}{2} \\ \frac{0}{36} & \frac{0}{36} & \frac{36}{36} \end{array}]$

Step 1.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 & 1 & - \frac{21}{2} \\ 0 & 0 & 1 \end{array}]$

Step 1.7

Perform the row operation $R_{2} = R_{2} + \frac{21}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 1.7.1

Perform the row operation $R_{2} = R_{2} + \frac{21}{2} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{5} & \frac{3}{5} \\ 0 + \frac{21}{2} \cdot 0 & 1 + \frac{21}{2} \cdot 0 & - \frac{21}{2} + \frac{21}{2} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 1.7.2

Simplify $R_{2}$ .

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

Step 1.8

Perform the row operation $R_{1} = R_{1} - \frac{3}{5} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 1.8.1

Perform the row operation $R_{1} = R_{1} - \frac{3}{5} R_{3}$ to make the entry at $1, 3$ a $0$ .

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

Step 1.8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{9}{5} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 1.9

Perform the row operation $R_{1} = R_{1} - \frac{9}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 1.9.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{5} \cdot 0 & \frac{9}{5} - \frac{9}{5} \cdot 1 & 0 - \frac{9}{5} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 1.9.2

Simplify $R_{1}$ .

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

Step 2

The pivot positions are the locations with the leading $1$ 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 $3$

Step 3

The rank is the number of pivot columns.

$3$