Linear Algebra Examples

S = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 - 1 342 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦, ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 62 - 1 34 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

$S = {[\begin{array}{c} 2 \\ - 1 \\ 3 \\ 4 \\ 2 \end{array}], [\begin{array}{c} 6 \\ 2 \\ - 1 \\ 3 \\ 4 \end{array}]}$

Step 1

Write as an augmented matrix for

A x = 0

$A x = 0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & 6 & 0 - 1 & 2 & 0 3 & - 1 & 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 2 & 6 & 0 \\ - 1 & 2 & 0 \\ 3 & - 1 & 0 \\ 4 & 3 & 0 \\ 2 & 4 & 0 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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 2.1.1

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{matrix} \frac{2}{2} & \frac{6}{2} & \frac{0}{2} - 1 & 2 & 0 3 & - 1 & 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{2}{2} & \frac{6}{2} & \frac{0}{2} \\ - 1 & 2 & 0 \\ 3 & - 1 & 0 \\ 4 & 3 & 0 \\ 2 & 4 & 0 \end{array}]$

Step 2.1.2

Simplify

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 3 & 0 - 1 & 2 & 0 3 & - 1 & 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 3 & 0 - 1 & 2 & 0 3 & - 1 & 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 2.2

Perform the row operation

R_{2} = R_{2} + R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

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Step 2.2.1

Perform the row operation

R_{2} = R_{2} + R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 3 & 0 - 1 + 1 \cdot 1 & 2 + 1 \cdot 3 & 0 + 0 3 & - 1 & 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 2.2.2

Simplify

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 3 & 0 0 & 5 & 0 3 & - 1 & 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 3 & - 1 & 0 \\ 4 & 3 & 0 \\ 2 & 4 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 3 & 0 0 & 5 & 0 3 & - 1 & 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 3 & - 1 & 0 \\ 4 & 3 & 0 \\ 2 & 4 & 0 \end{array}]$

Step 2.3

Perform the row operation

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

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

3, 1

$3, 1$ a

0

$0$ .

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Step 2.3.1

Perform the row operation

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

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

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 3 & 0 0 & 5 & 0 3 - 3 \cdot 1 & - 1 - 3 \cdot 3 & 0 - 3 \cdot 0 4 & 3 & 0 2 & 4 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 3 - 3 \cdot 1 & - 1 - 3 \cdot 3 & 0 - 3 \cdot 0 \\ 4 & 3 & 0 \\ 2 & 4 & 0 \end{array}]$

Step 2.3.2

Simplify

$R_{3}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 4 & 3 & 0 \\ 2 & 4 & 0 \end{array}]$

Step 2.4

Perform the row operation

$R_{4} = R_{4} - 4 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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Step 2.4.1

Perform the row operation

$R_{4} = R_{4} - 4 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 4 - 4 \cdot 1 & 3 - 4 \cdot 3 & 0 - 4 \cdot 0 \\ 2 & 4 & 0 \end{array}]$

Step 2.4.2

Simplify

$R_{4}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 2 & 4 & 0 \end{array}]$

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

$0$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 2 - 2 \cdot 1 & 4 - 2 \cdot 3 & 0 - 2 \cdot 0 \end{array}]$

Step 2.5.2

Simplify

$R_{5}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Step 2.6

Multiply each element of

$R_{2}$ by

$\frac{1}{5}$ to make the entry at

$2, 2$ a

$1$ .

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Step 2.6.1

Multiply each element of

$R_{2}$ by

$\frac{1}{5}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & 3 & 0 \\ \frac{0}{5} & \frac{5}{5} & \frac{0}{5} \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Step 2.6.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Step 2.7

Perform the row operation

$R_{3} = R_{3} + 10 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Step 2.7.1

Perform the row operation

$R_{3} = R_{3} + 10 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 + 10 \cdot 0 & - 10 + 10 \cdot 1 & 0 + 10 \cdot 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Step 2.7.2

Simplify

$R_{3}$ .

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

Step 2.8

Perform the row operation

$R_{4} = R_{4} + 9 R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Step 2.8.1

Perform the row operation

$R_{4} = R_{4} + 9 R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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

Step 2.8.2

Simplify

$R_{4}$ .

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

Step 2.9

Perform the row operation

$R_{5} = R_{5} + 2 R_{2}$ to make the entry at

$5, 2$ a

$0$ .

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Step 2.9.1

Perform the row operation

$R_{5} = R_{5} + 2 R_{2}$ to make the entry at

$5, 2$ a

$0$ .

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

Step 2.9.2

Simplify

$R_{5}$ .

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

Step 2.10

Perform the row operation

$R_{1} = R_{1} - 3 R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Step 2.10.1

Perform the row operation

$R_{1} = R_{1} - 3 R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

Step 2.10.2

Simplify

$R_{1}$ .

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

Step 3

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

$x = 0$

$y = 0$

$0 = 0$

Step 4

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

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

Step 5

Write as a solution set.

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

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