Linear Algebra Examples

$[\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}]$

Step 1

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 1.1

Perform the row operation

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

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 3 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 2 & 6 - 4 \cdot 3 \\ 7 & 8 & 9 \end{array}]$

Step 1.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 & 8 & 9 \end{array}]$

Step 2

Perform the row operation

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Step 2.1

Perform the row operation

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 - 7 \cdot 1 & 8 - 7 \cdot 2 & 9 - 7 \cdot 3 \end{array}]$

Step 2.2

Simplify

$R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 0 & - 6 & - 12 \end{array}]$

Step 3

Multiply each element of

$R_{2}$ by

$- \frac{1}{3}$ to make the entry at

$2, 2$ a

$1$ .

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Step 3.1

Multiply each element of

$R_{2}$ by

$- \frac{1}{3}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & 2 & 3 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 6 \\ 0 & - 6 & - 12 \end{array}]$

Step 3.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & - 6 & - 12 \end{array}]$

Step 4

Perform the row operation

$R_{3} = R_{3} + 6 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Step 4.1

Perform the row operation

$R_{3} = R_{3} + 6 R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 & - 12 + 6 \cdot 2 \end{array}]$

Step 4.2

Simplify

$R_{3}$ .

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

Step 5

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 5.1

Perform the row operation

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

$1, 2$ a

$0$ .

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

Step 5.2

Simplify

$R_{1}$ .

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