Linear Algebra Examples

$a + b + 2 d + g = 1$ , $a - b - d + g = 0$ , $b + d = - 1$ , $a + b + g = 2$

Step 1

Write the system of equations in matrix form.

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

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify $R_{2}$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify $R_{4}$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify $R_{2}$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify $R_{3}$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Simplify $R_{3}$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Simplify $R_{4}$ .

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

Step 2.7

Multiply each element of $R_{4}$ by $\frac{1}{7}$ to make the entry at $4, 5$ a $1$ .

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Step 2.7.1

Multiply each element of $R_{4}$ by $\frac{1}{7}$ to make the entry at $4, 5$ a $1$ .

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

Step 2.7.2

Simplify $R_{4}$ .

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

Step 2.8

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

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Step 2.8.1

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

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

Step 2.8.2

Simplify $R_{3}$ .

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

Step 2.9

Perform the row operation $R_{2} = R_{2} - \frac{1}{2} R_{4}$ to make the entry at $2, 5$ a $0$ .

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Step 2.9.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{2} R_{4}$ to make the entry at $2, 5$ a $0$ .

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

Step 2.9.2

Simplify $R_{2}$ .

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

Step 2.10

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

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Step 2.10.1

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

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

Step 2.10.2

Simplify $R_{1}$ .

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

Step 2.11

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

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Step 2.11.1

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

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

Step 2.11.2

Simplify $R_{2}$ .

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

Step 2.12

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

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Step 2.12.1

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

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

Step 2.12.2

Simplify $R_{1}$ .

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

Step 2.13

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

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Step 2.13.1

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

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

Step 2.13.2

Simplify $R_{1}$ .

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

Step 3

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

$a + g = 0$

$b = 0$

$d = 0$

$0 = 1$

Step 4

Subtract $g$ from both sides of the equation.

$a = - g$

$b = 0$

$d = 0$

$0 = 1$

Step 5

Since $0 \neq 1$ , there are no solutions.

No solution