Linear Algebra Examples

$S = {[\begin{array}{c} 2 \\ - 1 \\ 3 \\ 1 \\ 0 \\ 3 \end{array}], [\begin{array}{c} 9 \\ - 6 \\ - 2 \\ 3 \\ 2 \\ 1 \end{array}], [\begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \end{array}], [\begin{array}{c} - 3 \\ 1 \\ 4 \\ 2 \\ 1 \\ 2 \end{array}]}$

Step 1

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

$[\begin{array}{c} 2 & 9 & 1 & - 3 & 0 \\ - 1 & - 6 & 1 & 1 & 0 \\ 3 & - 2 & 1 & 4 & 0 \\ 1 & 3 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2

Find the reduced row echelon form.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{9}{2} & \frac{1}{2} & \frac{- 3}{2} & \frac{0}{2} \\ - 1 & - 6 & 1 & 1 & 0 \\ 3 & - 2 & 1 & 4 & 0 \\ 1 & 3 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ - 1 & - 6 & 1 & 1 & 0 \\ 3 & - 2 & 1 & 4 & 0 \\ 1 & 3 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ - 1 + 1 \cdot 1 & - 6 + \frac{9}{2} & 1 + \frac{1}{2} & 1 - \frac{3}{2} & 0 + 0 \\ 3 & - 2 & 1 & 4 & 0 \\ 1 & 3 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & 0 \\ 3 - 3 \cdot 1 & - 2 - 3 (\frac{9}{2}) & 1 - 3 (\frac{1}{2}) & 4 - 3 (- \frac{3}{2}) & 0 - 3 \cdot 0 \\ 1 & 3 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{31}{2} & - \frac{1}{2} & \frac{17}{2} & 0 \\ 1 & 3 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{31}{2} & - \frac{1}{2} & \frac{17}{2} & 0 \\ 1 - 1 & 3 - \frac{9}{2} & 0 - \frac{1}{2} & 2 + \frac{3}{2} & 0 - 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{3}{2} & \frac{3}{2} & - \frac{1}{2} & 0 \\ 0 & - \frac{31}{2} & - \frac{1}{2} & \frac{17}{2} & 0 \\ 0 & - \frac{3}{2} & - \frac{1}{2} & \frac{7}{2} & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 3 & 1 & 1 & 2 & 0 \end{array}]$

Step 2.5

Perform the row operation $R_{6} = R_{6} - 3 R_{1}$ to make the entry at $6, 1$ a $0$ .

Tap for more steps...

Step 2.5.1

Perform the row operation $R_{6} = R_{6} - 3 R_{1}$ to make the entry at $6, 1$ a $0$ .

Step 2.5.2

Simplify $R_{6}$ .

Step 2.6

Multiply each element of $R_{2}$ by $- \frac{2}{3}$ to make the entry at $2, 2$ a $1$ .

Tap for more steps...

Step 2.6.1

Multiply each element of $R_{2}$ by $- \frac{2}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ - \frac{2}{3} \cdot 0 & - \frac{2}{3} (- \frac{3}{2}) & - \frac{2}{3} \cdot \frac{3}{2} & - \frac{2}{3} (- \frac{1}{2}) & - \frac{2}{3} \cdot 0 \\ 0 & - \frac{31}{2} & - \frac{1}{2} & \frac{17}{2} & 0 \\ 0 & - \frac{3}{2} & - \frac{1}{2} & \frac{7}{2} & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.6.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & - \frac{31}{2} & - \frac{1}{2} & \frac{17}{2} & 0 \\ 0 & - \frac{3}{2} & - \frac{1}{2} & \frac{7}{2} & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.7

Perform the row operation $R_{3} = R_{3} + \frac{31}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

Tap for more steps...

Step 2.7.1

Perform the row operation $R_{3} = R_{3} + \frac{31}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 + \frac{31}{2} \cdot 0 & - \frac{31}{2} + \frac{31}{2} \cdot 1 & - \frac{1}{2} + \frac{31}{2} \cdot - 1 & \frac{17}{2} + \frac{31}{2} \cdot \frac{1}{3} & 0 + \frac{31}{2} \cdot 0 \\ 0 & - \frac{3}{2} & - \frac{1}{2} & \frac{7}{2} & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & - 16 & \frac{41}{3} & 0 \\ 0 & - \frac{3}{2} & - \frac{1}{2} & \frac{7}{2} & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.8

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

Tap for more steps...

Step 2.8.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & - 16 & \frac{41}{3} & 0 \\ 0 + \frac{3}{2} \cdot 0 & - \frac{3}{2} + \frac{3}{2} \cdot 1 & - \frac{1}{2} + \frac{3}{2} \cdot - 1 & \frac{7}{2} + \frac{3}{2} \cdot \frac{1}{3} & 0 + \frac{3}{2} \cdot 0 \\ 0 & 2 & 0 & 1 & 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & - 16 & \frac{41}{3} & 0 \\ 0 & 0 & - 2 & 4 & 0 \\ 0 & 2 & 0 & 1 & 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.9

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

Tap for more steps...

Step 2.9.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & - 16 & \frac{41}{3} & 0 \\ 0 & 0 & - 2 & 4 & 0 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot - 1 & 1 - 2 (\frac{1}{3}) & 0 - 2 \cdot 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.9.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & - 16 & \frac{41}{3} & 0 \\ 0 & 0 & - 2 & 4 & 0 \\ 0 & 0 & 2 & \frac{1}{3} & 0 \\ 0 & - \frac{25}{2} & - \frac{1}{2} & \frac{13}{2} & 0 \end{array}]$

Step 2.10

Perform the row operation $R_{6} = R_{6} + \frac{25}{2} R_{2}$ to make the entry at $6, 2$ a $0$ .

Tap for more steps...

Step 2.10.1

Perform the row operation $R_{6} = R_{6} + \frac{25}{2} R_{2}$ to make the entry at $6, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & - 16 & \frac{41}{3} & 0 \\ 0 & 0 & - 2 & 4 & 0 \\ 0 & 0 & 2 & \frac{1}{3} & 0 \\ 0 + \frac{25}{2} \cdot 0 & - \frac{25}{2} + \frac{25}{2} \cdot 1 & - \frac{1}{2} + \frac{25}{2} \cdot - 1 & \frac{13}{2} + \frac{25}{2} \cdot \frac{1}{3} & 0 + \frac{25}{2} \cdot 0 \end{array}]$

Step 2.10.2

Simplify $R_{6}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & - 16 & \frac{41}{3} & 0 \\ 0 & 0 & - 2 & 4 & 0 \\ 0 & 0 & 2 & \frac{1}{3} & 0 \\ 0 & 0 & - 13 & \frac{32}{3} & 0 \end{array}]$

Step 2.11

Multiply each element of $R_{3}$ by $- \frac{1}{16}$ to make the entry at $3, 3$ a $1$ .

Tap for more steps...

Step 2.11.1

Multiply each element of $R_{3}$ by $- \frac{1}{16}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ - \frac{1}{16} \cdot 0 & - \frac{1}{16} \cdot 0 & - \frac{1}{16} \cdot - 16 & - \frac{1}{16} \cdot \frac{41}{3} & - \frac{1}{16} \cdot 0 \\ 0 & 0 & - 2 & 4 & 0 \\ 0 & 0 & 2 & \frac{1}{3} & 0 \\ 0 & 0 & - 13 & \frac{32}{3} & 0 \end{array}]$

Step 2.11.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & - 2 & 4 & 0 \\ 0 & 0 & 2 & \frac{1}{3} & 0 \\ 0 & 0 & - 13 & \frac{32}{3} & 0 \end{array}]$

Step 2.12

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

Tap for more steps...

Step 2.12.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 4 + 2 (- \frac{41}{48}) & 0 + 2 \cdot 0 \\ 0 & 0 & 2 & \frac{1}{3} & 0 \\ 0 & 0 & - 13 & \frac{32}{3} & 0 \end{array}]$

Step 2.12.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & \frac{55}{24} & 0 \\ 0 & 0 & 2 & \frac{1}{3} & 0 \\ 0 & 0 & - 13 & \frac{32}{3} & 0 \end{array}]$

Step 2.13

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

Tap for more steps...

Step 2.13.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & \frac{55}{24} & 0 \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & \frac{1}{3} - 2 (- \frac{41}{48}) & 0 - 2 \cdot 0 \\ 0 & 0 & - 13 & \frac{32}{3} & 0 \end{array}]$

Step 2.13.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & \frac{55}{24} & 0 \\ 0 & 0 & 0 & \frac{49}{24} & 0 \\ 0 & 0 & - 13 & \frac{32}{3} & 0 \end{array}]$

Step 2.14

Perform the row operation $R_{6} = R_{6} + 13 R_{3}$ to make the entry at $6, 3$ a $0$ .

Tap for more steps...

Step 2.14.1

Perform the row operation $R_{6} = R_{6} + 13 R_{3}$ to make the entry at $6, 3$ a $0$ .

Step 2.14.2

Simplify $R_{6}$ .

Step 2.15

Multiply each element of $R_{4}$ by $\frac{24}{55}$ to make the entry at $4, 4$ a $1$ .

Tap for more steps...

Step 2.15.1

Multiply each element of $R_{4}$ by $\frac{24}{55}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ \frac{24}{55} \cdot 0 & \frac{24}{55} \cdot 0 & \frac{24}{55} \cdot 0 & \frac{24}{55} \cdot \frac{55}{24} & \frac{24}{55} \cdot 0 \\ 0 & 0 & 0 & \frac{49}{24} & 0 \\ 0 & 0 & 0 & - \frac{7}{16} & 0 \end{array}]$

Step 2.15.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{49}{24} & 0 \\ 0 & 0 & 0 & - \frac{7}{16} & 0 \end{array}]$

Step 2.16

Perform the row operation $R_{5} = R_{5} - \frac{49}{24} R_{4}$ to make the entry at $5, 4$ a $0$ .

Tap for more steps...

Step 2.16.1

Perform the row operation $R_{5} = R_{5} - \frac{49}{24} R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 - \frac{49}{24} \cdot 0 & 0 - \frac{49}{24} \cdot 0 & 0 - \frac{49}{24} \cdot 0 & \frac{49}{24} - \frac{49}{24} \cdot 1 & 0 - \frac{49}{24} \cdot 0 \\ 0 & 0 & 0 & - \frac{7}{16} & 0 \end{array}]$

Step 2.16.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{7}{16} & 0 \end{array}]$

Step 2.17

Perform the row operation $R_{6} = R_{6} + \frac{7}{16} R_{4}$ to make the entry at $6, 4$ a $0$ .

Tap for more steps...

Step 2.17.1

Perform the row operation $R_{6} = R_{6} + \frac{7}{16} R_{4}$ to make the entry at $6, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 + \frac{7}{16} \cdot 0 & 0 + \frac{7}{16} \cdot 0 & 0 + \frac{7}{16} \cdot 0 & - \frac{7}{16} + \frac{7}{16} \cdot 1 & 0 + \frac{7}{16} \cdot 0 \end{array}]$

Step 2.17.2

Simplify $R_{6}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 & 0 & 1 & - \frac{41}{48} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.18

Perform the row operation $R_{3} = R_{3} + \frac{41}{48} R_{4}$ to make the entry at $3, 4$ a $0$ .

Tap for more steps...

Step 2.18.1

Perform the row operation $R_{3} = R_{3} + \frac{41}{48} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & \frac{1}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - 1 & \frac{1}{3} & 0 \\ 0 + \frac{41}{48} \cdot 0 & 0 + \frac{41}{48} \cdot 0 & 1 + \frac{41}{48} \cdot 0 & - \frac{41}{48} + \frac{41}{48} \cdot 1 & 0 + \frac{41}{48} \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.18.2

Simplify $R_{3}$ .

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

Step 2.19

Perform the row operation $R_{2} = R_{2} - \frac{1}{3} R_{4}$ to make the entry at $2, 4$ a $0$ .

Tap for more steps...

Step 2.19.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{3} R_{4}$ to make the entry at $2, 4$ a $0$ .

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

Step 2.19.2

Simplify $R_{2}$ .

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

Step 2.20

Perform the row operation $R_{1} = R_{1} + \frac{3}{2} R_{4}$ to make the entry at $1, 4$ a $0$ .

Tap for more steps...

Step 2.20.1

Perform the row operation $R_{1} = R_{1} + \frac{3}{2} R_{4}$ to make the entry at $1, 4$ a $0$ .

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

Step 2.20.2

Simplify $R_{1}$ .

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

Step 2.21

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

Tap for more steps...

Step 2.21.1

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

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

Step 2.21.2

Simplify $R_{2}$ .

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

Step 2.22

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

Tap for more steps...

Step 2.22.1

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

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

Step 2.22.2

Simplify $R_{1}$ .

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

Step 2.23

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

Tap for more steps...

Step 2.23.1

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

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

Step 2.23.2

Simplify $R_{1}$ .

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

Step 3

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

$x_{1} = 0$

$x_{2} = 0$

$x_{3} = 0$

$x_{4} = 0$

$0 = 0$

Step 4

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

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]$

Step 5

Write as a solution set.

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

Enter YOUR Problem