Linear Algebra Examples

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

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

Step 1

Swap

R_{2}

$R_{2}$ with

R_{1}

$R_{1}$ to put a nonzero entry at

1, 1

$1, 1$ .

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

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

Step 2

Perform the row operation

R_{3} = R_{3} - 2 R_{1}

$R_{3} = R_{3} - 2 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Step 2.1

Perform the row operation

R_{3} = R_{3} - 2 R_{1}

$R_{3} = R_{3} - 2 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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

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

Step 2.2

Simplify

R_{3}

$R_{3}$ .

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

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

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

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

Step 3

Perform the row operation

R_{3} = R_{3} + R_{2}

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

3, 2

$3, 2$ a

0

$0$ .

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Step 3.1

Perform the row operation

R_{3} = R_{3} + R_{2}

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

3, 2

$3, 2$ a

0

$0$ .

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

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

Step 3.2

Simplify

R_{3}

$R_{3}$ .

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

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

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

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

Step 4

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

3, 4

$3, 4$ a

1

$1$ .

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Step 4.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

3, 4

$3, 4$ a

1

$1$ .

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

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

Step 4.2

Simplify

R_{3}

$R_{3}$ .

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

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

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

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

Step 5

Perform the row operation

R_{2} = R_{2} + 4 R_{3}

$R_{2} = R_{2} + 4 R_{3}$ to make the entry at

2, 4

$2, 4$ a

0

$0$ .

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Step 5.1

Perform the row operation

R_{2} = R_{2} + 4 R_{3}

$R_{2} = R_{2} + 4 R_{3}$ to make the entry at

2, 4

$2, 4$ a

0

$0$ .

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

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

Step 5.2

Simplify

R_{2}

$R_{2}$ .

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

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

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

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

Step 6

Perform the row operation

R_{1} = R_{1} + 2 R_{3}

$R_{1} = R_{1} + 2 R_{3}$ to make the entry at

1, 4

$1, 4$ a

0

$0$ .

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Step 6.1

Perform the row operation

R_{1} = R_{1} + 2 R_{3}

$R_{1} = R_{1} + 2 R_{3}$ to make the entry at

1, 4

$1, 4$ a

0

$0$ .

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

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

Step 6.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 4 & 3 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}]

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

[\begin{array}{c} 1 & 4 & 3 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}]

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

Step 7

Perform the row operation

R_{1} = R_{1} - 4 R_{2}

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

1, 2

$1, 2$ a

0

$0$ .

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Step 7.1

Perform the row operation

R_{1} = R_{1} - 4 R_{2}

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

1, 2

$1, 2$ a

0

$0$ .

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

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

Step 7.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 & - 17 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}]

$[\begin{array}{c} 1 & 0 & - 17 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

[\begin{array}{c} 1 & 0 & - 17 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}]

$[\begin{array}{c} 1 & 0 & - 17 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$