Linear Algebra Examples

$2 x_{1} + 4 x_{2} + 6 x_{3} = 18$ $4 x_{1} + 5 x_{2} - 6 x_{3} = 24$ $2 x_{1} + x_{2} - 12 x_{3} = 2$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 2 & 4 & 6 & 18 \\ 4 & 5 & - 6 & 24 \\ 2 & 1 & - 12 & 2 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{4}{2} & \frac{6}{2} & \frac{18}{2} \\ 4 & 5 & - 6 & 24 \\ 2 & 1 & - 12 & 2 \end{array}]$

Step 2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ 4 & 5 & - 6 & 24 \\ 2 & 1 & - 12 & 2 \end{array}]$

Step 2.2

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

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Step 2.2.1

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

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 2 & - 6 - 4 \cdot 3 & 24 - 4 \cdot 9 \\ 2 & 1 & - 12 & 2 \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ 0 & - 3 & - 18 & - 12 \\ 2 & 1 & - 12 & 2 \end{array}]$

Step 2.3

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

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Step 2.3.1

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

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ 0 & - 3 & - 18 & - 12 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 2 & - 12 - 2 \cdot 3 & 2 - 2 \cdot 9 \end{array}]$

Step 2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ 0 & - 3 & - 18 & - 12 \\ 0 & - 3 & - 18 & - 16 \end{array}]$

Step 2.4

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

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Step 2.4.1

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

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 18 & - \frac{1}{3} \cdot - 12 \\ 0 & - 3 & - 18 & - 16 \end{array}]$

Step 2.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ 0 & 1 & 6 & 4 \\ 0 & - 3 & - 18 & - 16 \end{array}]$

Step 2.5

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

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Step 2.5.1

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

$[\begin{array}{c} 1 & 2 & 3 & 9 \\ 0 & 1 & 6 & 4 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 18 + 3 \cdot 6 & - 16 + 3 \cdot 4 \end{array}]$

Step 2.5.2

Simplify $R_{3}$ .

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

Step 2.6

Multiply each element of $R_{3}$ by $- \frac{1}{4}$ to make the entry at $3, 4$ a $1$ .

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Step 2.6.1

Multiply each element of $R_{3}$ by $- \frac{1}{4}$ to make the entry at $3, 4$ a $1$ .

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

Step 2.6.2

Simplify $R_{3}$ .

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

Simplify $R_{2}$ .

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

Step 2.8

Perform the row operation $R_{1} = R_{1} - 9 R_{3}$ to make the entry at $1, 4$ a $0$ .

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Step 2.8.1

Perform the row operation $R_{1} = R_{1} - 9 R_{3}$ to make the entry at $1, 4$ a $0$ .

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

Step 2.8.2

Simplify $R_{1}$ .

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

Step 2.9

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 2.9.1

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

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

Step 2.9.2

Simplify $R_{1}$ .

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

Step 3

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

$x_{1} - 9 x_{3} = 0$

$x_{2} + 6 x_{3} = 0$

$0 = 1$

Step 4

The system is inconsistent so there is no solution.

$No solution$