Algebra Examples

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array}]$

Step 1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 1.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 6 - 6 \cdot 1 & 7 - 6 \cdot 2 & 8 - 6 \cdot 3 & 9 - 6 \cdot 4 & 10 - 6 \cdot 5 \\ 11 & 12 & 13 & 14 & 15 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array}]$

Step 1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & - 5 & - 10 & - 15 & - 20 \\ 11 & 12 & 13 & 14 & 15 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array}]$

Step 2

Perform the row operation $R_{3} = R_{3} - 11 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{3} = R_{3} - 11 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & - 5 & - 10 & - 15 & - 20 \\ 11 - 11 \cdot 1 & 12 - 11 \cdot 2 & 13 - 11 \cdot 3 & 14 - 11 \cdot 4 & 15 - 11 \cdot 5 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array}]$

Step 2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & - 5 & - 10 & - 15 & - 20 \\ 0 & - 10 & - 20 & - 30 & - 40 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array}]$

Step 3

Perform the row operation $R_{4} = R_{4} - 16 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 3.1

Perform the row operation $R_{4} = R_{4} - 16 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & - 5 & - 10 & - 15 & - 20 \\ 0 & - 10 & - 20 & - 30 & - 40 \\ 16 - 16 \cdot 1 & 17 - 16 \cdot 2 & 18 - 16 \cdot 3 & 19 - 16 \cdot 4 & 20 - 16 \cdot 5 \\ 21 & 22 & 23 & 24 & 25 \end{array}]$

Step 3.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & - 5 & - 10 & - 15 & - 20 \\ 0 & - 10 & - 20 & - 30 & - 40 \\ 0 & - 15 & - 30 & - 45 & - 60 \\ 21 & 22 & 23 & 24 & 25 \end{array}]$

Step 4

Perform the row operation $R_{5} = R_{5} - 21 R_{1}$ to make the entry at $5, 1$ a $0$ .

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Step 4.1

Perform the row operation $R_{5} = R_{5} - 21 R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & - 5 & - 10 & - 15 & - 20 \\ 0 & - 10 & - 20 & - 30 & - 40 \\ 0 & - 15 & - 30 & - 45 & - 60 \\ 21 - 21 \cdot 1 & 22 - 21 \cdot 2 & 23 - 21 \cdot 3 & 24 - 21 \cdot 4 & 25 - 21 \cdot 5 \end{array}]$

Step 4.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & - 5 & - 10 & - 15 & - 20 \\ 0 & - 10 & - 20 & - 30 & - 40 \\ 0 & - 15 & - 30 & - 45 & - 60 \\ 0 & - 20 & - 40 & - 60 & - 80 \end{array}]$

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 $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot - 10 & - \frac{1}{5} \cdot - 15 & - \frac{1}{5} \cdot - 20 \\ 0 & - 10 & - 20 & - 30 & - 40 \\ 0 & - 15 & - 30 & - 45 & - 60 \\ 0 & - 20 & - 40 & - 60 & - 80 \end{array}]$

Step 5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & - 10 & - 20 & - 30 & - 40 \\ 0 & - 15 & - 30 & - 45 & - 60 \\ 0 & - 20 & - 40 & - 60 & - 80 \end{array}]$

Step 6

Perform the row operation $R_{3} = R_{3} + 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 6.1

Perform the row operation $R_{3} = R_{3} + 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 + 10 \cdot 0 & - 10 + 10 \cdot 1 & - 20 + 10 \cdot 2 & - 30 + 10 \cdot 3 & - 40 + 10 \cdot 4 \\ 0 & - 15 & - 30 & - 45 & - 60 \\ 0 & - 20 & - 40 & - 60 & - 80 \end{array}]$

Step 6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & - 15 & - 30 & - 45 & - 60 \\ 0 & - 20 & - 40 & - 60 & - 80 \end{array}]$

Step 7

Perform the row operation $R_{4} = R_{4} + 15 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 7.1

Perform the row operation $R_{4} = R_{4} + 15 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 + 15 \cdot 0 & - 15 + 15 \cdot 1 & - 30 + 15 \cdot 2 & - 45 + 15 \cdot 3 & - 60 + 15 \cdot 4 \\ 0 & - 20 & - 40 & - 60 & - 80 \end{array}]$

Step 7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & - 20 & - 40 & - 60 & - 80 \end{array}]$

Step 8

Perform the row operation $R_{5} = R_{5} + 20 R_{2}$ to make the entry at $5, 2$ a $0$ .

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Step 8.1

Perform the row operation $R_{5} = R_{5} + 20 R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 + 20 \cdot 0 & - 20 + 20 \cdot 1 & - 40 + 20 \cdot 2 & - 60 + 20 \cdot 3 & - 80 + 20 \cdot 4 \end{array}]$

Step 8.2

Simplify $R_{5}$ .

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

Step 9

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 9.1

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

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

Step 9.2

Simplify $R_{1}$ .

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