Examples

$[\begin{array}{c} 3 & 6 \\ 8 & 9 \end{array}]$

Step 1

Write as an augmented matrix for

$A x = 0$ .

$[\begin{array}{c} 3 & 6 & 0 \\ 8 & 9 & 0 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

Multiply each element of

$R_{1}$ by

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

$1, 1$ a

$1$ .

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Step 2.1.1

Multiply each element of

$R_{1}$ by

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

$1, 1$ a

$1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{6}{3} & \frac{0}{3} \\ 8 & 9 & 0 \end{array}]$

Step 2.1.2

Simplify

$R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 \\ 8 & 9 & 0 \end{array}]$

Step 2.2

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 2.2.1

Perform the row operation

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

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 0 \\ 8 - 8 \cdot 1 & 9 - 8 \cdot 2 & 0 - 8 \cdot 0 \end{array}]$

Step 2.2.2

Simplify

$R_{2}$ .

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

Step 2.3

Multiply each element of

$R_{2}$ by

$- \frac{1}{7}$ to make the entry at

$2, 2$ a

$1$ .

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Step 2.3.1

Multiply each element of

$R_{2}$ by

$- \frac{1}{7}$ to make the entry at

$2, 2$ a

$1$ .

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

Step 2.3.2

Simplify

$R_{2}$ .

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

Step 2.4

Perform the row operation

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

$1, 2$ a

$0$ .

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Step 2.4.1

Perform the row operation

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

$1, 2$ a

$0$ .

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

Step 2.4.2

Simplify

$R_{1}$ .

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

Step 3

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

$x = 0$

$y = 0$

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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