Linear Algebra Examples

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

Step 1

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

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

Step 2

Find the reduced row echelon form.

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Step 2.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify $R_{1}$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify $R_{3}$ .

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

Step 2.4

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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Step 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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

Step 2.4.2

Simplify $R_{2}$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Simplify $R_{4}$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Simplify $R_{3}$ .

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

Simplify $R_{4}$ .

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

Step 2.8

Multiply each element of $R_{4}$ by $\frac{1}{11}$ to make the entry at $4, 4$ a $1$ .

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Step 2.8.1

Multiply each element of $R_{4}$ by $\frac{1}{11}$ to make the entry at $4, 4$ a $1$ .

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

Step 2.8.2

Simplify $R_{4}$ .

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

Step 2.9

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

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Step 2.9.1

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

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

Step 2.9.2

Simplify $R_{3}$ .

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

Step 2.10

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

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Step 2.10.1

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

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

Step 2.10.2

Simplify $R_{2}$ .

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

Step 2.11

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

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Step 2.11.1

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

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

Step 2.11.2

Simplify $R_{1}$ .

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

Step 2.12

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

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Step 2.12.1

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

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

Step 2.12.2

Simplify $R_{2}$ .

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

Step 2.13

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

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Step 2.13.1

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

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

Step 2.13.2

Simplify $R_{1}$ .

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

Step 2.14

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

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Step 2.14.1

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

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

Step 2.14.2

Simplify $R_{1}$ .

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

Step 3

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

$x_{1} = 0$

$x_{2} = 0$

$x_{3} = 0$

$x_{4} = 0$

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.

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