Linear Algebra Examples

$S = {[\begin{array}{c} 2 \\ - 1 \\ 3 \\ 1 \\ 2 \end{array}], [\begin{array}{c} 1 \\ 2 \\ - 1 \\ 5 \\ 2 \end{array}], [\begin{array}{c} 2 \\ 1 \\ - 3 \\ 6 \\ 1 \end{array}]}$

Step 1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 2 & 1 & 2 & 0 \\ - 1 & 2 & 1 & 0 \\ 3 & - 1 & - 3 & 0 \\ 1 & 5 & 6 & 0 \\ 2 & 2 & 1 & 0 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify $R_{1}$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify $R_{2}$ .

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

Step 2.3

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 2.3.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & \frac{5}{2} & 2 & 0 \\ 3 - 3 \cdot 1 & - 1 - 3 (\frac{1}{2}) & - 3 - 3 \cdot 1 & 0 - 3 \cdot 0 \\ 1 & 5 & 6 & 0 \\ 2 & 2 & 1 & 0 \end{array}]$

Step 2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & \frac{5}{2} & 2 & 0 \\ 0 & - \frac{5}{2} & - 6 & 0 \\ 1 & 5 & 6 & 0 \\ 2 & 2 & 1 & 0 \end{array}]$

Step 2.4

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

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Step 2.4.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & \frac{5}{2} & 2 & 0 \\ 0 & - \frac{5}{2} & - 6 & 0 \\ 1 - 1 & 5 - \frac{1}{2} & 6 - 1 & 0 - 0 \\ 2 & 2 & 1 & 0 \end{array}]$

Step 2.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & \frac{5}{2} & 2 & 0 \\ 0 & - \frac{5}{2} & - 6 & 0 \\ 0 & \frac{9}{2} & 5 & 0 \\ 2 & 2 & 1 & 0 \end{array}]$

Step 2.5

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

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Step 2.5.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & \frac{5}{2} & 2 & 0 \\ 0 & - \frac{5}{2} & - 6 & 0 \\ 0 & \frac{9}{2} & 5 & 0 \\ 2 - 2 \cdot 1 & 2 - 2 (\frac{1}{2}) & 1 - 2 \cdot 1 & 0 - 2 \cdot 0 \end{array}]$

Step 2.5.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & \frac{5}{2} & 2 & 0 \\ 0 & - \frac{5}{2} & - 6 & 0 \\ 0 & \frac{9}{2} & 5 & 0 \\ 0 & 1 & - 1 & 0 \end{array}]$

Step 2.6

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

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Step 2.6.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ \frac{2}{5} \cdot 0 & \frac{2}{5} \cdot \frac{5}{2} & \frac{2}{5} \cdot 2 & \frac{2}{5} \cdot 0 \\ 0 & - \frac{5}{2} & - 6 & 0 \\ 0 & \frac{9}{2} & 5 & 0 \\ 0 & 1 & - 1 & 0 \end{array}]$

Step 2.6.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & - \frac{5}{2} & - 6 & 0 \\ 0 & \frac{9}{2} & 5 & 0 \\ 0 & 1 & - 1 & 0 \end{array}]$

Step 2.7

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

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Step 2.7.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 + \frac{5}{2} \cdot 0 & - \frac{5}{2} + \frac{5}{2} \cdot 1 & - 6 + \frac{5}{2} \cdot \frac{4}{5} & 0 + \frac{5}{2} \cdot 0 \\ 0 & \frac{9}{2} & 5 & 0 \\ 0 & 1 & - 1 & 0 \end{array}]$

Step 2.7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & - 4 & 0 \\ 0 & \frac{9}{2} & 5 & 0 \\ 0 & 1 & - 1 & 0 \end{array}]$

Step 2.8

Perform the row operation $R_{4} = R_{4} - \frac{9}{2} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 2.8.1

Perform the row operation $R_{4} = R_{4} - \frac{9}{2} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & - 4 & 0 \\ 0 - \frac{9}{2} \cdot 0 & \frac{9}{2} - \frac{9}{2} \cdot 1 & 5 - \frac{9}{2} \cdot \frac{4}{5} & 0 - \frac{9}{2} \cdot 0 \\ 0 & 1 & - 1 & 0 \end{array}]$

Step 2.8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & - 4 & 0 \\ 0 & 0 & \frac{7}{5} & 0 \\ 0 & 1 & - 1 & 0 \end{array}]$

Step 2.9

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

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Step 2.9.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & - 4 & 0 \\ 0 & 0 & \frac{7}{5} & 0 \\ 0 - 0 & 1 - 1 & - 1 - \frac{4}{5} & 0 - 0 \end{array}]$

Step 2.9.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & - 4 & 0 \\ 0 & 0 & \frac{7}{5} & 0 \\ 0 & 0 & - \frac{9}{5} & 0 \end{array}]$

Step 2.10

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

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Step 2.10.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot - 4 & - \frac{1}{4} \cdot 0 \\ 0 & 0 & \frac{7}{5} & 0 \\ 0 & 0 & - \frac{9}{5} & 0 \end{array}]$

Step 2.10.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & \frac{7}{5} & 0 \\ 0 & 0 & - \frac{9}{5} & 0 \end{array}]$

Step 2.11

Perform the row operation $R_{4} = R_{4} - \frac{7}{5} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 2.11.1

Perform the row operation $R_{4} = R_{4} - \frac{7}{5} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & 1 & 0 \\ 0 - \frac{7}{5} \cdot 0 & 0 - \frac{7}{5} \cdot 0 & \frac{7}{5} - \frac{7}{5} \cdot 1 & 0 - \frac{7}{5} \cdot 0 \\ 0 & 0 & - \frac{9}{5} & 0 \end{array}]$

Step 2.11.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{9}{5} & 0 \end{array}]$

Step 2.12

Perform the row operation $R_{5} = R_{5} + \frac{9}{5} R_{3}$ to make the entry at $5, 3$ a $0$ .

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Step 2.12.1

Perform the row operation $R_{5} = R_{5} + \frac{9}{5} R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 + \frac{9}{5} \cdot 0 & 0 + \frac{9}{5} \cdot 0 & - \frac{9}{5} + \frac{9}{5} \cdot 1 & 0 + \frac{9}{5} \cdot 0 \end{array}]$

Step 2.12.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 & 1 & \frac{4}{5} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.13

Perform the row operation $R_{2} = R_{2} - \frac{4}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 2.13.1

Perform the row operation $R_{2} = R_{2} - \frac{4}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 1 & 0 \\ 0 - \frac{4}{5} \cdot 0 & 1 - \frac{4}{5} \cdot 0 & \frac{4}{5} - \frac{4}{5} \cdot 1 & 0 - \frac{4}{5} \cdot 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 2.13.2

Simplify $R_{2}$ .

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

Step 2.14

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

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Step 2.14.1

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

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

Step 2.14.2

Simplify $R_{1}$ .

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

Step 2.15

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 2.15.1

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

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

Step 2.15.2

Simplify $R_{1}$ .

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

Step 3

Use the result matrix to declare the final solution to the system of equations.

$x = 0$

$y = 0$

$z = 0$

$0 = 0$

Step 4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 5

Write as a solution set.

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

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