Linear Algebra Examples

$[\begin{array}{c} 2 & 0 & 6 \\ - 3 & 5 & 2 \\ 4 & 1 & 9 \\ 1 & - 2 & 11 \end{array}]$

Step 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 1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{0}{2} & \frac{6}{2} \\ - 3 & 5 & 2 \\ 4 & 1 & 9 \\ 1 & - 2 & 11 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

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

Step 2

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

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Step 2.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ - 3 + 3 \cdot 1 & 5 + 3 \cdot 0 & 2 + 3 \cdot 3 \\ 4 & 1 & 9 \\ 1 & - 2 & 11 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

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

Step 3

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

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Step 3.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 5 & 11 \\ 4 - 4 \cdot 1 & 1 - 4 \cdot 0 & 9 - 4 \cdot 3 \\ 1 & - 2 & 11 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 5 & 11 \\ 0 & 1 & - 3 \\ 0 & - 2 & 8 \end{array}]$

Step 5

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Step 5.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & 3 \\ \frac{0}{5} & \frac{5}{5} & \frac{11}{5} \\ 0 & 1 & - 3 \\ 0 & - 2 & 8 \end{array}]$

Step 5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 & 1 & - 3 \\ 0 & - 2 & 8 \end{array}]$

Step 6

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

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Step 6.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 - 0 & 1 - 1 & - 3 - \frac{11}{5} \\ 0 & - 2 & 8 \end{array}]$

Step 6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 & 0 & - \frac{26}{5} \\ 0 & - 2 & 8 \end{array}]$

Step 7

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 7.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 & 0 & - \frac{26}{5} \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 8 + 2 (\frac{11}{5}) \end{array}]$

Step 7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 & 0 & - \frac{26}{5} \\ 0 & 0 & \frac{62}{5} \end{array}]$

Step 8

Multiply each element of $R_{3}$ by $- \frac{5}{26}$ to make the entry at $3, 3$ a $1$ .

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Step 8.1

Multiply each element of $R_{3}$ by $- \frac{5}{26}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ - \frac{5}{26} \cdot 0 & - \frac{5}{26} \cdot 0 & - \frac{5}{26} (- \frac{26}{5}) \\ 0 & 0 & \frac{62}{5} \end{array}]$

Step 8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 & 0 & 1 \\ 0 & 0 & \frac{62}{5} \end{array}]$

Step 9

Perform the row operation $R_{4} = R_{4} - \frac{62}{5} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 9.1

Perform the row operation $R_{4} = R_{4} - \frac{62}{5} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 & 0 & 1 \\ 0 - \frac{62}{5} \cdot 0 & 0 - \frac{62}{5} \cdot 0 & \frac{62}{5} - \frac{62}{5} \cdot 1 \end{array}]$

Step 9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 1 & \frac{11}{5} \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}]$

Step 10

Perform the row operation $R_{2} = R_{2} - \frac{11}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 10.1

Perform the row operation $R_{2} = R_{2} - \frac{11}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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

Step 10.2

Simplify $R_{2}$ .

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

Step 11

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 11.1

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

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

Step 11.2

Simplify $R_{1}$ .

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