Find Reduced Row Echelon Form [[10,4,20,2.95],[7.5,6,15,3.77],[15,3,40,1.8]]

$[\begin{array}{c} 10 & 4 & 20 & 2.95 \\ 7.5 & 6 & 15 & 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 1

Perform the row operation $R_{1} = \frac{1}{10} R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $1$ .

Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = \frac{1}{10} R_{1}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} \frac{1}{10} R_{1} & \frac{1}{10} R_{1} & \frac{1}{10} R_{1} & \frac{1}{10} R_{1} \\ 7.5 & 6 & 15 & 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

$R_{1} = \frac{1}{10} R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = \frac{1}{10} R_{1}$ .

$[\begin{array}{c} (\frac{1}{10}) \cdot (10) & (\frac{1}{10}) \cdot (4) & (\frac{1}{10}) \cdot (20) & (\frac{1}{10}) \cdot (2.95) \\ 7.5 & 6 & 15 & 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

$R_{1} = \frac{1}{10} R_{1}$

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

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 7.5 & 6 & 15 & 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 2

Perform the row operation $R_{2} = - 7.5 \cdot R_{1} + R_{2}$ 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 $R_{2} = - 7.5 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ - 7.5 \cdot R_{1} + R_{2} & - 7.5 \cdot R_{1} + R_{2} & - 7.5 \cdot R_{1} + R_{2} & - 7.5 \cdot R_{1} + R_{2} \\ 15 & 3 & 40 & 1.8 \end{array}]$

$R_{2} = - 7.5 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 7.5 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ (- 7.5) \cdot (1) + 7.5 & (- 7.5) \cdot (\frac{2}{5}) + 6 & (- 7.5) \cdot (2) + 15 & (- 7.5) \cdot (0.295) + 3.77 \\ 15 & 3 & 40 & 1.8 \end{array}]$

$R_{2} = - 7.5 \cdot R_{1} + R_{2}$

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

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 3 & 0 & 1.5575 \\ 15 & 3 & 40 & 1.8 \end{array}]$

Step 3

Perform the row operation $R_{3} = - 15 \cdot R_{1} + R_{3}$ 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 $R_{3} = - 15 \cdot R_{1} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 3 & 0 & 1.5575 \\ - 15 \cdot R_{1} + R_{3} & - 15 \cdot R_{1} + R_{3} & - 15 \cdot R_{1} + R_{3} & - 15 \cdot R_{1} + R_{3} \end{array}]$

$R_{3} = - 15 \cdot R_{1} + R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - 15 \cdot R_{1} + R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 3 & 0 & 1.5575 \\ (- 15) \cdot (1) + 15 & (- 15) \cdot (\frac{2}{5}) + 3 & (- 15) \cdot (2) + 40 & (- 15) \cdot (0.295) + 1.8 \end{array}]$

$R_{3} = - 15 \cdot R_{1} + R_{3}$

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

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 3 & 0 & 1.5575 \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

Step 4

Perform the row operation $R_{2} = 0.\overline{3} \cdot R_{2}$ 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 $R_{2} = 0.\overline{3} \cdot R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0.\overline{3} \cdot R_{2} & 0.\overline{3} \cdot R_{2} & 0.\overline{3} \cdot R_{2} & 0.\overline{3} \cdot R_{2} \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

$R_{2} = 0.\overline{3} \cdot R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = 0.\overline{3} \cdot R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ (0.\overline{3}) \cdot (0) & (0.\overline{3}) \cdot (3) & (0.\overline{3}) \cdot (0) & (0.\overline{3}) \cdot (1.5575) \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

$R_{2} = 0.\overline{3} \cdot R_{2}$

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

$[\begin{array}{c} 1 & \frac{2}{5} & 2 & 0.295 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

Step 5

Perform the row operation $R_{1} = - 0.4 \cdot R_{2} + R_{1}$ 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 $R_{1} = - 0.4 \cdot R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - 0.4 \cdot R_{2} + R_{1} & - 0.4 \cdot R_{2} + R_{1} & - 0.4 \cdot R_{2} + R_{1} & - 0.4 \cdot R_{2} + R_{1} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

$R_{1} = - 0.4 \cdot R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - 0.4 \cdot R_{2} + R_{1}$ .

$[\begin{array}{c} (- 0.4) \cdot (0) + 1 & (- 0.4) \cdot (1) + \frac{2}{5} & (- 0.4) \cdot (0) + 2 & (- 0.4) \cdot (0.5191\overline{6}) + 0.295 \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

$R_{1} = - 0.4 \cdot R_{2} + R_{1}$

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

$[\begin{array}{c} 1 & 0 & 2 & 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & - 3 & 10 & - 2.625 \end{array}]$

Step 6

Perform the row operation $R_{3} = 3 \cdot R_{2} + R_{3}$ 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 $R_{3} = 3 \cdot R_{2} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & 2 & 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 3 \cdot R_{2} + R_{3} & 3 \cdot R_{2} + R_{3} & 3 \cdot R_{2} + R_{3} & 3 \cdot R_{2} + R_{3} \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & 2 & 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ (3) \cdot (0) + 0 & (3) \cdot (1) - 3 & (3) \cdot (0) + 10 & (3) \cdot (0.5191\overline{6}) - 2.625 \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & 2 & 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 10 & - 1.0675 \end{array}]$

Step 7

Perform the row operation $R_{3} = 0.1 \cdot R_{3}$ 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 $R_{3} = 0.1 \cdot R_{3}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & 2 & 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0.1 \cdot R_{3} & 0.1 \cdot R_{3} & 0.1 \cdot R_{3} & 0.1 \cdot R_{3} \end{array}]$

$R_{3} = 0.1 \cdot R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = 0.1 \cdot R_{3}$ .

$[\begin{array}{c} 1 & 0 & 2 & 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ (0.1) \cdot (0) & (0.1) \cdot (0) & (0.1) \cdot (10) & (0.1) \cdot (- 1.0675) \end{array}]$

$R_{3} = 0.1 \cdot R_{3}$

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

$[\begin{array}{c} 1 & 0 & 2 & 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

Step 8

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

$[\begin{array}{c} - 2 \cdot R_{3} + R_{1} & - 2 \cdot R_{3} + R_{1} & - 2 \cdot R_{3} + R_{1} & - 2 \cdot R_{3} + R_{1} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

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

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

$[\begin{array}{c} (- 2) \cdot (0) + 1 & (- 2) \cdot (0) + 0 & (- 2) \cdot (1) + 2 & (- 2) \cdot (- 0.10675) + 0.087\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & 0 & 0.3008\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

Step 9

Perform the row operation $R_{2} = 1 \cdot R_{2}$ 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 $R_{2} = 1 \cdot R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0.3008\overline{3} \\ 1 \cdot R_{2} & 1 \cdot R_{2} & 1 \cdot R_{2} & 1 \cdot R_{2} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

$R_{2} = 1 \cdot R_{2}$

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

$[\begin{array}{c} 1 & 0 & 0 & 0.3008\overline{3} \\ (1) \cdot (0) & (1) \cdot (1) & (1) \cdot (0) & (1) \cdot (0.5191\overline{6}) \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

$R_{2} = 1 \cdot R_{2}$

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

$[\begin{array}{c} 1 & 0 & 0 & 0.3008\overline{3} \\ 0 & 1 & 0 & 0.5191\overline{6} \\ 0 & 0 & 1 & - 0.10675 \end{array}]$

Find Reduced Row Echelon Form [[10,4,20,2.95],[7.5,6,15,3.77],[15,3,40,1.8]]

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