Linear Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 4 & 7 \\ 0 & 1 & 0 & - 8 \\ 0 & 0 & 1 & 5 \end{array}]$

Step 2

Perform the row operation $R_{1} = R_{1} - 4 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 2.1

Perform the row operation $R_{1} = R_{1} - 4 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 4 \cdot 0 & - 3 - 4 \cdot 0 & 4 - 4 \cdot 1 & 7 - 4 \cdot 5 \\ 0 & 1 & 0 & - 8 \\ 0 & 0 & 1 & 5 \end{array}]$

Step 2.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 3 & 0 & - 13 \\ 0 & 1 & 0 & - 8 \\ 0 & 0 & 1 & 5 \end{array}]$

Step 3

Perform the row operation $R_{1} = R_{1} + 3 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.1

Perform the row operation $R_{1} = R_{1} + 3 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 & - 13 + 3 \cdot - 8 \\ 0 & 1 & 0 & - 8 \\ 0 & 0 & 1 & 5 \end{array}]$

Step 3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 37 \\ 0 & 1 & 0 & - 8 \\ 0 & 0 & 1 & 5 \end{array}]$