Linear Algebra Examples

$x + 3 y - 3 z - w = 11$ , $4 x + y + 2 z + 5 w = 3$ , $- 3 x - y + z - 3 w = - 7$ , $z - y - 3 z - 5 w = - 1$

Step 1

Subtract $3 z$ from $z$ .

$x + 3 y - 3 z - w = 11, 4 x + y + 2 z + 5 w = 3, - 3 x - y + z - 3 w = - 7, - y - 2 z - 5 w = - 1$

Step 2

Write the system of equations in matrix form.

$[\begin{array}{c} - 1 & 1 & 3 & - 3 & 11 \\ 5 & 4 & 1 & 2 & 3 \\ - 3 & - 3 & - 1 & 1 & - 7 \\ - 5 & 0 & - 1 & - 2 & - 1 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

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

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Step 3.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 1 & - 1 \cdot 3 & - - 3 & - 1 \cdot 11 \\ 5 & 4 & 1 & 2 & 3 \\ - 3 & - 3 & - 1 & 1 & - 7 \\ - 5 & 0 & - 1 & - 2 & - 1 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 5 & 4 & 1 & 2 & 3 \\ - 3 & - 3 & - 1 & 1 & - 7 \\ - 5 & 0 & - 1 & - 2 & - 1 \end{array}]$

Step 3.2

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

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Step 3.2.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 5 - 5 \cdot 1 & 4 - 5 \cdot - 1 & 1 - 5 \cdot - 3 & 2 - 5 \cdot 3 & 3 - 5 \cdot - 11 \\ - 3 & - 3 & - 1 & 1 & - 7 \\ - 5 & 0 & - 1 & - 2 & - 1 \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 9 & 16 & - 13 & 58 \\ - 3 & - 3 & - 1 & 1 & - 7 \\ - 5 & 0 & - 1 & - 2 & - 1 \end{array}]$

Step 3.3

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 3.3.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 9 & 16 & - 13 & 58 \\ - 3 + 3 \cdot 1 & - 3 + 3 \cdot - 1 & - 1 + 3 \cdot - 3 & 1 + 3 \cdot 3 & - 7 + 3 \cdot - 11 \\ - 5 & 0 & - 1 & - 2 & - 1 \end{array}]$

Step 3.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 9 & 16 & - 13 & 58 \\ 0 & - 6 & - 10 & 10 & - 40 \\ - 5 & 0 & - 1 & - 2 & - 1 \end{array}]$

Step 3.4

Perform the row operation $R_{4} = R_{4} + 5 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 3.4.1

Perform the row operation $R_{4} = R_{4} + 5 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 9 & 16 & - 13 & 58 \\ 0 & - 6 & - 10 & 10 & - 40 \\ - 5 + 5 \cdot 1 & 0 + 5 \cdot - 1 & - 1 + 5 \cdot - 3 & - 2 + 5 \cdot 3 & - 1 + 5 \cdot - 11 \end{array}]$

Step 3.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 9 & 16 & - 13 & 58 \\ 0 & - 6 & - 10 & 10 & - 40 \\ 0 & - 5 & - 16 & 13 & - 56 \end{array}]$

Step 3.5

Multiply each element of $R_{2}$ by $\frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

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Step 3.5.1

Multiply each element of $R_{2}$ by $\frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ \frac{0}{9} & \frac{9}{9} & \frac{16}{9} & \frac{- 13}{9} & \frac{58}{9} \\ 0 & - 6 & - 10 & 10 & - 40 \\ 0 & - 5 & - 16 & 13 & - 56 \end{array}]$

Step 3.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & - 6 & - 10 & 10 & - 40 \\ 0 & - 5 & - 16 & 13 & - 56 \end{array}]$

Step 3.6

Perform the row operation $R_{3} = R_{3} + 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 3.6.1

Perform the row operation $R_{3} = R_{3} + 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 & - 10 + 6 (\frac{16}{9}) & 10 + 6 (- \frac{13}{9}) & - 40 + 6 (\frac{58}{9}) \\ 0 & - 5 & - 16 & 13 & - 56 \end{array}]$

Step 3.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & \frac{2}{3} & \frac{4}{3} & - \frac{4}{3} \\ 0 & - 5 & - 16 & 13 & - 56 \end{array}]$

Step 3.7

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

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Step 3.7.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & \frac{2}{3} & \frac{4}{3} & - \frac{4}{3} \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 16 + 5 (\frac{16}{9}) & 13 + 5 (- \frac{13}{9}) & - 56 + 5 (\frac{58}{9}) \end{array}]$

Step 3.7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & \frac{2}{3} & \frac{4}{3} & - \frac{4}{3} \\ 0 & 0 & - \frac{64}{9} & \frac{52}{9} & - \frac{214}{9} \end{array}]$

Step 3.8

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

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Step 3.8.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ \frac{3}{2} \cdot 0 & \frac{3}{2} \cdot 0 & \frac{3}{2} \cdot \frac{2}{3} & \frac{3}{2} \cdot \frac{4}{3} & \frac{3}{2} (- \frac{4}{3}) \\ 0 & 0 & - \frac{64}{9} & \frac{52}{9} & - \frac{214}{9} \end{array}]$

Step 3.8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & 1 & 2 & - 2 \\ 0 & 0 & - \frac{64}{9} & \frac{52}{9} & - \frac{214}{9} \end{array}]$

Step 3.9

Perform the row operation $R_{4} = R_{4} + \frac{64}{9} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 3.9.1

Perform the row operation $R_{4} = R_{4} + \frac{64}{9} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & 1 & 2 & - 2 \\ 0 + \frac{64}{9} \cdot 0 & 0 + \frac{64}{9} \cdot 0 & - \frac{64}{9} + \frac{64}{9} \cdot 1 & \frac{52}{9} + \frac{64}{9} \cdot 2 & - \frac{214}{9} + \frac{64}{9} \cdot - 2 \end{array}]$

Step 3.9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & 1 & 2 & - 2 \\ 0 & 0 & 0 & 20 & - 38 \end{array}]$

Step 3.10

Multiply each element of $R_{4}$ by $\frac{1}{20}$ to make the entry at $4, 4$ a $1$ .

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Step 3.10.1

Multiply each element of $R_{4}$ by $\frac{1}{20}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & 1 & 2 & - 2 \\ \frac{0}{20} & \frac{0}{20} & \frac{0}{20} & \frac{20}{20} & \frac{- 38}{20} \end{array}]$

Step 3.10.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & 1 & 2 & - 2 \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.11

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

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Step 3.11.1

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

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 2 - 2 (- \frac{19}{10}) \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.11.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & - \frac{13}{9} & \frac{58}{9} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.12

Perform the row operation $R_{2} = R_{2} + \frac{13}{9} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 3.12.1

Perform the row operation $R_{2} = R_{2} + \frac{13}{9} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 + \frac{13}{9} \cdot 0 & 1 + \frac{13}{9} \cdot 0 & \frac{16}{9} + \frac{13}{9} \cdot 0 & - \frac{13}{9} + \frac{13}{9} \cdot 1 & \frac{58}{9} + \frac{13}{9} (- \frac{19}{10}) \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.12.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 3 & - 11 \\ 0 & 1 & \frac{16}{9} & 0 & \frac{37}{10} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.13

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

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Step 3.13.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & - 1 - 3 \cdot 0 & - 3 - 3 \cdot 0 & 3 - 3 \cdot 1 & - 11 - 3 (- \frac{19}{10}) \\ 0 & 1 & \frac{16}{9} & 0 & \frac{37}{10} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.13.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 0 & - \frac{53}{10} \\ 0 & 1 & \frac{16}{9} & 0 & \frac{37}{10} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.14

Perform the row operation $R_{2} = R_{2} - \frac{16}{9} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 3.14.1

Perform the row operation $R_{2} = R_{2} - \frac{16}{9} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 0 & - \frac{53}{10} \\ 0 - \frac{16}{9} \cdot 0 & 1 - \frac{16}{9} \cdot 0 & \frac{16}{9} - \frac{16}{9} \cdot 1 & 0 - \frac{16}{9} \cdot 0 & \frac{37}{10} - \frac{16}{9} \cdot \frac{9}{5} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.14.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 3 & 0 & - \frac{53}{10} \\ 0 & 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.15

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

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Step 3.15.1

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

$[\begin{array}{c} 1 + 3 \cdot 0 & - 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 & - \frac{53}{10} + 3 (\frac{9}{5}) \\ 0 & 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.15.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 & \frac{1}{10} \\ 0 & 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.16

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

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Step 3.16.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + 0 & \frac{1}{10} + \frac{1}{2} \\ 0 & 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 3.16.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & \frac{3}{5} \\ 0 & 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & \frac{9}{5} \\ 0 & 0 & 0 & 1 & - \frac{19}{10} \end{array}]$

Step 4

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

$w = \frac{3}{5}$

$x = \frac{1}{2}$

$y = \frac{9}{5}$

$z = - \frac{19}{10}$

Step 5

The solution is the set of ordered pairs that makes the system true.

$(\frac{3}{5}, \frac{1}{2}, \frac{9}{5}, - \frac{19}{10})$

Step 6

Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.

$X = [\begin{array}{c} w \\ x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{3}{5} \\ \frac{1}{2} \\ \frac{9}{5} \\ - \frac{19}{10} \end{array}]$