Find Reduced Row Echelon Form [[-27,15,-19,-14],[3,-4,-21,20],[-2,16,-9,-18]]

$[\begin{array}{c} - 27 & 15 & - 19 & - 14 \\ 3 & - 4 & - 21 & 20 \\ - 2 & 16 & - 9 & - 18 \end{array}]$

Step 1

Perform the row operation $R_{1} = - \frac{1}{27} R_{1}$ 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 $R_{1} = - \frac{1}{27} R_{1}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} - \frac{1}{27} R_{1} & - \frac{1}{27} R_{1} & - \frac{1}{27} R_{1} & - \frac{1}{27} R_{1} \\ 3 & - 4 & - 21 & 20 \\ - 2 & 16 & - 9 & - 18 \end{array}]$

$R_{1} = - \frac{1}{27} R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - \frac{1}{27} R_{1}$ .

$[\begin{array}{c} (- \frac{1}{27}) \cdot (- 27) & (- \frac{1}{27}) \cdot (15) & (- \frac{1}{27}) \cdot (- 19) & (- \frac{1}{27}) \cdot (- 14) \\ 3 & - 4 & - 21 & 20 \\ - 2 & 16 & - 9 & - 18 \end{array}]$

$R_{1} = - \frac{1}{27} R_{1}$

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

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ 3 & - 4 & - 21 & 20 \\ - 2 & 16 & - 9 & - 18 \end{array}]$

Step 2

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 $0$ .

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} & - 3 \cdot R_{1} + R_{2} \\ - 2 & 16 & - 9 & - 18 \end{array}]$

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

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

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ (- 3) \cdot (1) + 3 & (- 3) \cdot (- \frac{5}{9}) - 4 & (- 3) \cdot (\frac{19}{27}) - 21 & (- 3) \cdot (\frac{14}{27}) + 20 \\ - 2 & 16 & - 9 & - 18 \end{array}]$

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

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

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ 0 & - \frac{7}{3} & - \frac{208}{9} & \frac{166}{9} \\ - 2 & 16 & - 9 & - 18 \end{array}]$

Step 3

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

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ 0 & - \frac{7}{3} & - \frac{208}{9} & \frac{166}{9} \\ 2 \cdot R_{1} + R_{3} & 2 \cdot R_{1} + R_{3} & 2 \cdot R_{1} + R_{3} & 2 \cdot R_{1} + R_{3} \end{array}]$

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

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

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ 0 & - \frac{7}{3} & - \frac{208}{9} & \frac{166}{9} \\ (2) \cdot (1) - 2 & (2) \cdot (- \frac{5}{9}) + 16 & (2) \cdot (\frac{19}{27}) - 9 & (2) \cdot (\frac{14}{27}) - 18 \end{array}]$

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

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

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ 0 & - \frac{7}{3} & - \frac{208}{9} & \frac{166}{9} \\ 0 & \frac{134}{9} & - \frac{205}{27} & - \frac{458}{27} \end{array}]$

Step 4

Perform the row operation $R_{2} = - \frac{3}{7} R_{2}$ 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 $R_{2} = - \frac{3}{7} R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ - \frac{3}{7} R_{2} & - \frac{3}{7} R_{2} & - \frac{3}{7} R_{2} & - \frac{3}{7} R_{2} \\ 0 & \frac{134}{9} & - \frac{205}{27} & - \frac{458}{27} \end{array}]$

$R_{2} = - \frac{3}{7} R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - \frac{3}{7} R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ (- \frac{3}{7}) \cdot (0) & (- \frac{3}{7}) \cdot (- \frac{7}{3}) & (- \frac{3}{7}) \cdot (- \frac{208}{9}) & (- \frac{3}{7}) \cdot (\frac{166}{9}) \\ 0 & \frac{134}{9} & - \frac{205}{27} & - \frac{458}{27} \end{array}]$

$R_{2} = - \frac{3}{7} R_{2}$

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

$[\begin{array}{c} 1 & - \frac{5}{9} & \frac{19}{27} & \frac{14}{27} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & \frac{134}{9} & - \frac{205}{27} & - \frac{458}{27} \end{array}]$

Step 5

Perform the row operation $R_{1} = \frac{5}{9} R_{2} + R_{1}$ 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 $R_{1} = \frac{5}{9} R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} \frac{5}{9} R_{2} + R_{1} & \frac{5}{9} R_{2} + R_{1} & \frac{5}{9} R_{2} + R_{1} & \frac{5}{9} R_{2} + R_{1} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & \frac{134}{9} & - \frac{205}{27} & - \frac{458}{27} \end{array}]$

$R_{1} = \frac{5}{9} R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = \frac{5}{9} R_{2} + R_{1}$ .

$[\begin{array}{c} (\frac{5}{9}) \cdot (0) + 1 & (\frac{5}{9}) \cdot (1) - \frac{5}{9} & (\frac{5}{9}) \cdot (\frac{208}{21}) + \frac{19}{27} & (\frac{5}{9}) \cdot (- \frac{166}{21}) + \frac{14}{27} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & \frac{134}{9} & - \frac{205}{27} & - \frac{458}{27} \end{array}]$

$R_{1} = \frac{5}{9} R_{2} + R_{1}$

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

$[\begin{array}{c} 1 & 0 & \frac{391}{63} & - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & \frac{134}{9} & - \frac{205}{27} & - \frac{458}{27} \end{array}]$

Step 6

Perform the row operation $R_{3} = - \frac{134}{9} R_{2} + R_{3}$ 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 $R_{3} = - \frac{134}{9} R_{2} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & \frac{391}{63} & - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ - \frac{134}{9} R_{2} + R_{3} & - \frac{134}{9} R_{2} + R_{3} & - \frac{134}{9} R_{2} + R_{3} & - \frac{134}{9} R_{2} + R_{3} \end{array}]$

$R_{3} = - \frac{134}{9} R_{2} + R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - \frac{134}{9} R_{2} + R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{391}{63} & - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ (- \frac{134}{9}) \cdot (0) + 0 & (- \frac{134}{9}) \cdot (1) + \frac{134}{9} & (- \frac{134}{9}) \cdot (\frac{208}{21}) - \frac{205}{27} & (- \frac{134}{9}) \cdot (- \frac{166}{21}) - \frac{458}{27} \end{array}]$

$R_{3} = - \frac{134}{9} R_{2} + R_{3}$

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

$[\begin{array}{c} 1 & 0 & \frac{391}{63} & - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & 0 & - \frac{9769}{63} & \frac{6346}{63} \end{array}]$

Step 7

Perform the row operation $R_{3} = - \frac{63}{9769} R_{3}$ 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 $R_{3} = - \frac{63}{9769} R_{3}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & \frac{391}{63} & - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ - \frac{63}{9769} R_{3} & - \frac{63}{9769} R_{3} & - \frac{63}{9769} R_{3} & - \frac{63}{9769} R_{3} \end{array}]$

$R_{3} = - \frac{63}{9769} R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - \frac{63}{9769} R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{391}{63} & - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ (- \frac{63}{9769}) \cdot (0) & (- \frac{63}{9769}) \cdot (0) & (- \frac{63}{9769}) \cdot (- \frac{9769}{63}) & (- \frac{63}{9769}) \cdot (\frac{6346}{63}) \end{array}]$

$R_{3} = - \frac{63}{9769} R_{3}$

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

$[\begin{array}{c} 1 & 0 & \frac{391}{63} & - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & 0 & 1 & - \frac{6346}{9769} \end{array}]$

Step 8

Perform the row operation $R_{1} = - \frac{391}{63} R_{3} + R_{1}$ 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 $R_{1} = - \frac{391}{63} R_{3} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - \frac{391}{63} R_{3} + R_{1} & - \frac{391}{63} R_{3} + R_{1} & - \frac{391}{63} R_{3} + R_{1} & - \frac{391}{63} R_{3} + R_{1} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & 0 & 1 & - \frac{6346}{9769} \end{array}]$

$R_{1} = - \frac{391}{63} R_{3} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - \frac{391}{63} R_{3} + R_{1}$ .

$[\begin{array}{c} (- \frac{391}{63}) \cdot (0) + 1 & (- \frac{391}{63}) \cdot (0) + 0 & (- \frac{391}{63}) \cdot (1) + \frac{391}{63} & (- \frac{391}{63}) \cdot (- \frac{6346}{9769}) - \frac{244}{63} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & 0 & 1 & - \frac{6346}{9769} \end{array}]$

$R_{1} = - \frac{391}{63} R_{3} + R_{1}$

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

$[\begin{array}{c} 1 & 0 & 0 & \frac{1550}{9769} \\ 0 & 1 & \frac{208}{21} & - \frac{166}{21} \\ 0 & 0 & 1 & - \frac{6346}{9769} \end{array}]$

Step 9

Perform the row operation $R_{2} = - \frac{208}{21} R_{3} + R_{2}$ 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 $R_{2} = - \frac{208}{21} R_{3} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{1550}{9769} \\ - \frac{208}{21} R_{3} + R_{2} & - \frac{208}{21} R_{3} + R_{2} & - \frac{208}{21} R_{3} + R_{2} & - \frac{208}{21} R_{3} + R_{2} \\ 0 & 0 & 1 & - \frac{6346}{9769} \end{array}]$

$R_{2} = - \frac{208}{21} R_{3} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - \frac{208}{21} R_{3} + R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{1550}{9769} \\ (- \frac{208}{21}) \cdot (0) + 0 & (- \frac{208}{21}) \cdot (0) + 1 & (- \frac{208}{21}) \cdot (1) + \frac{208}{21} & (- \frac{208}{21}) \cdot (- \frac{6346}{9769}) - \frac{166}{21} \\ 0 & 0 & 1 & - \frac{6346}{9769} \end{array}]$

$R_{2} = - \frac{208}{21} R_{3} + R_{2}$

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

$[\begin{array}{c} 1 & 0 & 0 & \frac{1550}{9769} \\ 0 & 1 & 0 & - \frac{14366}{9769} \\ 0 & 0 & 1 & - \frac{6346}{9769} \end{array}]$

Find Reduced Row Echelon Form [[-27,15,-19,-14],[3,-4,-21,20],[-2,16,-9,-18]]

Related Questions

Support LCMGCF.com by disabling your adblocker.