Linear Algebra Examples

rref $[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 1 & 0 & 0 & 1 & 6 \\ 0 & 0 & 1 & 0 & 6 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 1 - 1 & 0 - 1 & 0 - 1 & 1 - 0 & 6 - 5 \\ 0 & 0 & 1 & 0 & 6 \end{array}]$

Step 1.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & - 1 & - 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 6 \end{array}]$

Step 2

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

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Step 2.1

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

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 + 0 & - 1 + 1 \cdot 1 & - 1 + 0 & 1 + 0 & 1 + 1 \cdot 5 \\ 0 & 0 & 1 & 0 & 6 \end{array}]$

Step 2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & - 1 & 1 & 6 \\ 0 & 0 & 1 & 0 & 6 \end{array}]$

Step 3

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

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Step 3.1

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

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ - 0 & - 0 & - - 1 & - 1 \cdot 1 & - 1 \cdot 6 \\ 0 & 0 & 1 & 0 & 6 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & - 1 & - 6 \\ 0 & 0 & 1 & 0 & 6 \end{array}]$

Step 4

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

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Step 4.1

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

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & - 1 & - 6 \\ 0 - 0 & 0 - 0 & 1 - 1 & 0 + 1 & 6 + 6 \end{array}]$

Step 4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & - 1 & - 6 \\ 0 & 0 & 0 & 1 & 12 \end{array}]$

Step 5

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

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Step 5.1

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

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 + 0 & 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & - 6 + 1 \cdot 12 \\ 0 & 0 & 0 & 1 & 12 \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 12 \end{array}]$

Step 6

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

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Step 6.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 0 - 0 & 5 - 6 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 12 \end{array}]$

Step 6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 0 & - 1 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 12 \end{array}]$

Step 7

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

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Step 7.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 0 - 0 & - 1 - 5 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 12 \end{array}]$

Step 7.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & - 6 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 12 \end{array}]$