Find Reduced Row Echelon Form [[1,-1,4,-13],[3,0,1,-2],[1,3,1,13]]

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ 3 & 0 & 1 & - 2 \\ 1 & 3 & 1 & 13 \end{array}]$

Step 1

Perform the row operation $R_{2} = - 3 \cdot R_{1} + R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $0$ .

Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - 3 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} \\ 1 & 3 & 1 & 13 \end{array}]$

$R_{2} = - 3 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 3 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ (- 3) \cdot (1) + 3 & (- 3) \cdot (- 1) + 0 & (- 3) \cdot (4) + 1 & (- 3) \cdot (- 13) - 2 \\ 1 & 3 & 1 & 13 \end{array}]$

$R_{2} = - 3 \cdot R_{1} + R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ 0 & 3 & - 11 & 37 \\ 1 & 3 & 1 & 13 \end{array}]$

Step 2

Perform the row operation $R_{3} = - 1 \cdot R_{1} + R_{3}$ on $R_{3}$ (row $3$ ) in order to convert some elements in the row to $0$ .

Replace $R_{3}$ (row $3$ ) with the row operation $R_{3} = - 1 \cdot R_{1} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ 0 & 3 & - 11 & 37 \\ - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} \end{array}]$

$R_{3} = - 1 \cdot R_{1} + R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - 1 \cdot R_{1} + R_{3}$ .

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ 0 & 3 & - 11 & 37 \\ (- 1) \cdot (1) + 1 & (- 1) \cdot (- 1) + 3 & (- 1) \cdot (4) + 1 & (- 1) \cdot (- 13) + 13 \end{array}]$

$R_{3} = - 1 \cdot R_{1} + R_{3}$

Simplify $R_{3}$ (row $3$ ).

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ 0 & 3 & - 11 & 37 \\ 0 & 4 & - 3 & 26 \end{array}]$

Step 3

Perform the row operation $R_{2} = \frac{1}{3} R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $1$ .

Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = \frac{1}{3} R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ \frac{1}{3} R_{2} & \frac{1}{3} R_{2} & \frac{1}{3} R_{2} & \frac{1}{3} R_{2} \\ 0 & 4 & - 3 & 26 \end{array}]$

$R_{2} = \frac{1}{3} R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = \frac{1}{3} R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ (\frac{1}{3}) \cdot (0) & (\frac{1}{3}) \cdot (3) & (\frac{1}{3}) \cdot (- 11) & (\frac{1}{3}) \cdot (37) \\ 0 & 4 & - 3 & 26 \end{array}]$

$R_{2} = \frac{1}{3} R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & - 1 & 4 & - 13 \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 4 & - 3 & 26 \end{array}]$

Step 4

Perform the row operation $R_{1} = R_{2} + R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $0$ .

Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} R_{2} + R_{1} & R_{2} + R_{1} & R_{2} + R_{1} & R_{2} + R_{1} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 4 & - 3 & 26 \end{array}]$

$R_{1} = R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = R_{2} + R_{1}$ .

$[\begin{array}{c} 0 + 1 & 1 - 1 & - \frac{11}{3} + 4 & \frac{37}{3} - 13 \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 4 & - 3 & 26 \end{array}]$

$R_{1} = R_{2} + R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 4 & - 3 & 26 \end{array}]$

Step 5

Perform the row operation $R_{3} = - 4 \cdot R_{2} + R_{3}$ on $R_{3}$ (row $3$ ) in order to convert some elements in the row to $0$ .

Replace $R_{3}$ (row $3$ ) with the row operation $R_{3} = - 4 \cdot R_{2} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ - 4 \cdot R_{2} + R_{3} & - 4 \cdot R_{2} + R_{3} & - 4 \cdot R_{2} + R_{3} & - 4 \cdot R_{2} + R_{3} \end{array}]$

$R_{3} = - 4 \cdot R_{2} + R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - 4 \cdot R_{2} + R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ (- 4) \cdot (0) + 0 & (- 4) \cdot (1) + 4 & (- 4) \cdot (- \frac{11}{3}) - 3 & (- 4) \cdot (\frac{37}{3}) + 26 \end{array}]$

$R_{3} = - 4 \cdot R_{2} + R_{3}$

Simplify $R_{3}$ (row $3$ ).

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 0 & \frac{35}{3} & - \frac{70}{3} \end{array}]$

Step 6

Perform the row operation $R_{3} = \frac{3}{35} R_{3}$ on $R_{3}$ (row $3$ ) in order to convert some elements in the row to $1$ .

Replace $R_{3}$ (row $3$ ) with the row operation $R_{3} = \frac{3}{35} R_{3}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ \frac{3}{35} R_{3} & \frac{3}{35} R_{3} & \frac{3}{35} R_{3} & \frac{3}{35} R_{3} \end{array}]$

$R_{3} = \frac{3}{35} R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = \frac{3}{35} R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ (\frac{3}{35}) \cdot (0) & (\frac{3}{35}) \cdot (0) & (\frac{3}{35}) \cdot (\frac{35}{3}) & (\frac{3}{35}) \cdot (- \frac{70}{3}) \end{array}]$

$R_{3} = \frac{3}{35} R_{3}$

Simplify $R_{3}$ (row $3$ ).

$[\begin{array}{c} 1 & 0 & \frac{1}{3} & - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 0 & 1 & - 2 \end{array}]$

Step 7

Perform the row operation $R_{1} = - \frac{1}{3} R_{3} + R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $0$ .

Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = - \frac{1}{3} R_{3} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - \frac{1}{3} R_{3} + R_{1} & - \frac{1}{3} R_{3} + R_{1} & - \frac{1}{3} R_{3} + R_{1} & - \frac{1}{3} R_{3} + R_{1} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 0 & 1 & - 2 \end{array}]$

$R_{1} = - \frac{1}{3} R_{3} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - \frac{1}{3} R_{3} + R_{1}$ .

$[\begin{array}{c} (- \frac{1}{3}) \cdot (0) + 1 & (- \frac{1}{3}) \cdot (0) + 0 & (- \frac{1}{3}) \cdot (1) + \frac{1}{3} & (- \frac{1}{3}) \cdot (- 2) - \frac{2}{3} \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 0 & 1 & - 2 \end{array}]$

$R_{1} = - \frac{1}{3} R_{3} + R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & - \frac{11}{3} & \frac{37}{3} \\ 0 & 0 & 1 & - 2 \end{array}]$

Step 8

Perform the row operation $R_{2} = \frac{11}{3} R_{3} + R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $0$ .

Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = \frac{11}{3} R_{3} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ \frac{11}{3} R_{3} + R_{2} & \frac{11}{3} R_{3} + R_{2} & \frac{11}{3} R_{3} + R_{2} & \frac{11}{3} R_{3} + R_{2} \\ 0 & 0 & 1 & - 2 \end{array}]$

$R_{2} = \frac{11}{3} R_{3} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = \frac{11}{3} R_{3} + R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ (\frac{11}{3}) \cdot (0) + 0 & (\frac{11}{3}) \cdot (0) + 1 & (\frac{11}{3}) \cdot (1) - \frac{11}{3} & (\frac{11}{3}) \cdot (- 2) + \frac{37}{3} \\ 0 & 0 & 1 & - 2 \end{array}]$

$R_{2} = \frac{11}{3} R_{3} + R_{2}$

Simplify $R_{2}$ (row $2$ ).

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

Find Reduced Row Echelon Form [[1,-1,4,-13],[3,0,1,-2],[1,3,1,13]]

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