Find Reduced Row Echelon Form [[1,1,1,1],[1,-1,3,-1],[1,1,-1,1]]

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

Step 1

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

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

Simplify $R_{2}$ .

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

Step 2

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

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

Simplify $R_{3}$ .

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

Step 3

Multiply each element of $R_{2}$ by $- \frac{1}{2}$ to make the entry at $2, 2$ a $1$ .

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

Simplify $R_{2}$ .

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

Step 4

Multiply each element of $R_{3}$ by $- \frac{1}{2}$ to make the entry at $3, 3$ a $1$ .

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

Simplify $R_{3}$ .

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

Step 5

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

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

Simplify $R_{2}$ .

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

Step 6

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

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

Simplify $R_{1}$ .

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

Step 7

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

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

Simplify $R_{1}$ .

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

Find Reduced Row Echelon Form [[1,1,1,1],[1,-1,3,-1],[1,1,-1,1]]

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