Linear Algebra Examples

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 0 1 & 0 & 1 1 & 0 & 1 2 & 1 & 0 2 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 1

Perform the row operation

R_{2} = R_{2} - R_{1}

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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Step 1.1

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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

Step 1.2

Simplify

$R_{2}$ .

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

Step 2

Perform the row operation

$R_{3} = R_{3} - R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Step 2.1

Perform the row operation

$R_{3} = R_{3} - R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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

Step 2.2

Simplify

$R_{3}$ .

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

Step 3

Perform the row operation

$R_{4} = R_{4} - 2 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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Step 3.1

Perform the row operation

$R_{4} = R_{4} - 2 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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

Step 3.2

Simplify

$R_{4}$ .

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

Step 4

Perform the row operation

$R_{5} = R_{5} - 2 R_{1}$ to make the entry at

$5, 1$ a

$0$ .

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Step 4.1

Perform the row operation

$R_{5} = R_{5} - 2 R_{1}$ to make the entry at

$5, 1$ a

$0$ .

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

Step 4.2

Simplify

$R_{5}$ .

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

Step 5

Multiply each element of

$R_{2}$ by

$- 1$ to make the entry at

$2, 2$ a

$1$ .

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Step 5.1

Multiply each element of

$R_{2}$ by

$- 1$ to make the entry at

$2, 2$ a

$1$ .

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

Step 5.2

Simplify

$R_{2}$ .

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

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

$0$ .

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

Step 6.2

Simplify

$R_{3}$ .

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

Step 7

Perform the row operation

$R_{4} = R_{4} + R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Step 7.1

Perform the row operation

$R_{4} = R_{4} + R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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

Step 7.2

Simplify

$R_{4}$ .

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

Step 8

Perform the row operation

$R_{5} = R_{5} + R_{2}$ to make the entry at

$5, 2$ a

$0$ .

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Step 8.1

Perform the row operation

$R_{5} = R_{5} + R_{2}$ to make the entry at

$5, 2$ a

$0$ .

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

Step 8.2

Simplify

$R_{5}$ .

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

Step 9

Swap

$R_{4}$ with

$R_{3}$ to put a nonzero entry at

$3, 3$ .

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

Step 10

Multiply each element of

$R_{3}$ by

$- 1$ to make the entry at

$3, 3$ a

$1$ .

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Step 10.1

Multiply each element of

$R_{3}$ by

$- 1$ to make the entry at

$3, 3$ a

$1$ .

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

Step 10.2

Simplify

$R_{3}$ .

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

Step 11

Perform the row operation

$R_{5} = R_{5} + R_{3}$ to make the entry at

$5, 3$ a

$0$ .

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Step 11.1

Perform the row operation

$R_{5} = R_{5} + R_{3}$ to make the entry at

$5, 3$ a

$0$ .

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

Step 11.2

Simplify

$R_{5}$ .

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

Step 12

Perform the row operation

$R_{2} = R_{2} + R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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Step 12.1

Perform the row operation

$R_{2} = R_{2} + R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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

Step 12.2

Simplify

$R_{2}$ .

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

Step 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 13.1

Perform the row operation

$R_{1} = R_{1} - R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

Step 13.2

Simplify

$R_{1}$ .

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