Find Reduced Row Echelon Form [[-2,0,4,0],[167,0,-119,-14],[15,70,-15,0],[-325,0,225,100]]

$[\begin{array}{c} - 2 & 0 & 4 & 0 \\ 167 & 0 & - 119 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

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

$[\begin{array}{c} - \frac{1}{2} R_{1} & - \frac{1}{2} R_{1} & - \frac{1}{2} R_{1} & - \frac{1}{2} R_{1} \\ 167 & 0 & - 119 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

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

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

$[\begin{array}{c} (- \frac{1}{2}) \cdot (- 2) & (- \frac{1}{2}) \cdot (0) & (- \frac{1}{2}) \cdot (4) & (- \frac{1}{2}) \cdot (0) \\ 167 & 0 & - 119 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 167 & 0 & - 119 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Perform the row operation $R_{2} = - 167 \cdot R_{1} + R_{2}$ 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 $R_{2} = - 167 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ - 167 \cdot R_{1} + R_{2} & - 167 \cdot R_{1} + R_{2} & - 167 \cdot R_{1} + R_{2} & - 167 \cdot R_{1} + R_{2} \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

$R_{2} = - 167 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 167 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ (- 167) \cdot (1) + 167 & (- 167) \cdot (0) + 0 & (- 167) \cdot (- 2) - 119 & (- 167) \cdot (0) - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

$R_{2} = - 167 \cdot R_{1} + R_{2}$

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

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 15 & 70 & - 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ - 15 \cdot R_{1} + R_{3} & - 15 \cdot R_{1} + R_{3} & - 15 \cdot R_{1} + R_{3} & - 15 \cdot R_{1} + R_{3} \\ - 325 & 0 & 225 & 100 \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ (- 15) \cdot (1) + 15 & (- 15) \cdot (0) + 70 & (- 15) \cdot (- 2) - 15 & (- 15) \cdot (0) + 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 70 & 15 & 0 \\ - 325 & 0 & 225 & 100 \end{array}]$

Perform the row operation $R_{4} = 325 \cdot R_{1} + R_{4}$ on $R_{4}$ (row $4$ ) in order to convert some elements in the row to $0$ .

Replace $R_{4}$ (row $4$ ) with the row operation $R_{4} = 325 \cdot R_{1} + R_{4}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 70 & 15 & 0 \\ 325 \cdot R_{1} + R_{4} & 325 \cdot R_{1} + R_{4} & 325 \cdot R_{1} + R_{4} & 325 \cdot R_{1} + R_{4} \end{array}]$

$R_{4} = 325 \cdot R_{1} + R_{4}$

Replace $R_{4}$ (row $4$ ) with the actual values of the elements for the row operation $R_{4} = 325 \cdot R_{1} + R_{4}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 70 & 15 & 0 \\ (325) \cdot (1) - 325 & (325) \cdot (0) + 0 & (325) \cdot (- 2) + 225 & (325) \cdot (0) + 100 \end{array}]$

$R_{4} = 325 \cdot R_{1} + R_{4}$

Simplify $R_{4}$ (row $4$ ).

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 70 & 15 & 0 \\ 0 & 0 & - 425 & 100 \end{array}]$

Exchange row $3$ and row $2$ to organize the zeros into position.

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 70 & 15 & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 0 & - 425 & 100 \end{array}]$

$R_{3} \leftrightarrow R$

Perform the row operation $R_{2} = \frac{1}{70} R_{2}$ 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 $R_{2} = \frac{1}{70} R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ \frac{1}{70} R_{2} & \frac{1}{70} R_{2} & \frac{1}{70} R_{2} & \frac{1}{70} R_{2} \\ 0 & 0 & 215 & - 14 \\ 0 & 0 & - 425 & 100 \end{array}]$

$R_{2} = \frac{1}{70} R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = \frac{1}{70} R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ (\frac{1}{70}) \cdot (0) & (\frac{1}{70}) \cdot (70) & (\frac{1}{70}) \cdot (15) & (\frac{1}{70}) \cdot (0) \\ 0 & 0 & 215 & - 14 \\ 0 & 0 & - 425 & 100 \end{array}]$

$R_{2} = \frac{1}{70} R_{2}$

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

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 215 & - 14 \\ 0 & 0 & - 425 & 100 \end{array}]$

Perform the row operation $R_{3} = \frac{1}{215} R_{3}$ 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 $R_{3} = \frac{1}{215} R_{3}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ \frac{1}{215} R_{3} & \frac{1}{215} R_{3} & \frac{1}{215} R_{3} & \frac{1}{215} R_{3} \\ 0 & 0 & - 425 & 100 \end{array}]$

$R_{3} = \frac{1}{215} R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = \frac{1}{215} R_{3}$ .

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ (\frac{1}{215}) \cdot (0) & (\frac{1}{215}) \cdot (0) & (\frac{1}{215}) \cdot (215) & (\frac{1}{215}) \cdot (- 14) \\ 0 & 0 & - 425 & 100 \end{array}]$

$R_{3} = \frac{1}{215} R_{3}$

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

$[\begin{array}{c} 1 & 0 & - 2 & 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

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 $0$ .

$[\begin{array}{c} 2 \cdot R_{3} + R_{1} & 2 \cdot R_{3} + R_{1} & 2 \cdot R_{3} + R_{1} & 2 \cdot R_{3} + R_{1} \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

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

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

$[\begin{array}{c} (2) \cdot (0) + 1 & (2) \cdot (0) + 0 & (2) \cdot (1) - 2 & (2) \cdot (- \frac{14}{215}) + 0 \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & \frac{3}{14} & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

Perform the row operation $R_{2} = - \frac{3}{14} R_{3} + R_{2}$ 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 $R_{2} = - \frac{3}{14} R_{3} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ - \frac{3}{14} R_{3} + R_{2} & - \frac{3}{14} R_{3} + R_{2} & - \frac{3}{14} R_{3} + R_{2} & - \frac{3}{14} R_{3} + R_{2} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

$R_{2} = - \frac{3}{14} R_{3} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - \frac{3}{14} R_{3} + R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ (- \frac{3}{14}) \cdot (0) + 0 & (- \frac{3}{14}) \cdot (0) + 1 & (- \frac{3}{14}) \cdot (1) + \frac{3}{14} & (- \frac{3}{14}) \cdot (- \frac{14}{215}) + 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

$R_{2} = - \frac{3}{14} R_{3} + R_{2}$

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

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & - 425 & 100 \end{array}]$

Perform the row operation $R_{4} = 425 \cdot R_{3} + R_{4}$ on $R_{4}$ (row $4$ ) in order to convert some elements in the row to $0$ .

Replace $R_{4}$ (row $4$ ) with the row operation $R_{4} = 425 \cdot R_{3} + R_{4}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 425 \cdot R_{3} + R_{4} & 425 \cdot R_{3} + R_{4} & 425 \cdot R_{3} + R_{4} & 425 \cdot R_{3} + R_{4} \end{array}]$

$R_{4} = 425 \cdot R_{3} + R_{4}$

Replace $R_{4}$ (row $4$ ) with the actual values of the elements for the row operation $R_{4} = 425 \cdot R_{3} + R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ (425) \cdot (0) + 0 & (425) \cdot (0) + 0 & (425) \cdot (1) - 425 & (425) \cdot (- \frac{14}{215}) + 100 \end{array}]$

$R_{4} = 425 \cdot R_{3} + R_{4}$

Simplify $R_{4}$ (row $4$ ).

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & \frac{3110}{43} \end{array}]$

Perform the row operation $R_{4} = \frac{43}{3110} R_{4}$ on $R_{4}$ (row $4$ ) in order to convert some elements in the row to $1$ .

Replace $R_{4}$ (row $4$ ) with the row operation $R_{4} = \frac{43}{3110} R_{4}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ \frac{43}{3110} R_{4} & \frac{43}{3110} R_{4} & \frac{43}{3110} R_{4} & \frac{43}{3110} R_{4} \end{array}]$

$R_{4} = \frac{43}{3110} R_{4}$

Replace $R_{4}$ (row $4$ ) with the actual values of the elements for the row operation $R_{4} = \frac{43}{3110} R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ (\frac{43}{3110}) \cdot (0) & (\frac{43}{3110}) \cdot (0) & (\frac{43}{3110}) \cdot (0) & (\frac{43}{3110}) \cdot (\frac{3110}{43}) \end{array}]$

$R_{4} = \frac{43}{3110} R_{4}$

Simplify $R_{4}$ (row $4$ ).

$[\begin{array}{c} 1 & 0 & 0 & - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{1} = \frac{28}{215} R_{4} + R_{1}$ 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 $R_{1} = \frac{28}{215} R_{4} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} \frac{28}{215} R_{4} + R_{1} & \frac{28}{215} R_{4} + R_{1} & \frac{28}{215} R_{4} + R_{1} & \frac{28}{215} R_{4} + R_{1} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

$R_{1} = \frac{28}{215} R_{4} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = \frac{28}{215} R_{4} + R_{1}$ .

$[\begin{array}{c} (\frac{28}{215}) \cdot (0) + 1 & (\frac{28}{215}) \cdot (0) + 0 & (\frac{28}{215}) \cdot (0) + 0 & (\frac{28}{215}) \cdot (1) - \frac{28}{215} \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

$R_{1} = \frac{28}{215} R_{4} + R_{1}$

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

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{2} = - \frac{3}{215} R_{4} + R_{2}$ 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 $R_{2} = - \frac{3}{215} R_{4} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ - \frac{3}{215} R_{4} + R_{2} & - \frac{3}{215} R_{4} + R_{2} & - \frac{3}{215} R_{4} + R_{2} & - \frac{3}{215} R_{4} + R_{2} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

$R_{2} = - \frac{3}{215} R_{4} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - \frac{3}{215} R_{4} + R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ (- \frac{3}{215}) \cdot (0) + 0 & (- \frac{3}{215}) \cdot (0) + 1 & (- \frac{3}{215}) \cdot (0) + 0 & (- \frac{3}{215}) \cdot (1) + \frac{3}{215} \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

$R_{2} = - \frac{3}{215} R_{4} + R_{2}$

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

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{3} = \frac{14}{215} R_{4} + R_{3}$ 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 $R_{3} = \frac{14}{215} R_{4} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \frac{14}{215} R_{4} + R_{3} & \frac{14}{215} R_{4} + R_{3} & \frac{14}{215} R_{4} + R_{3} & \frac{14}{215} R_{4} + R_{3} \\ 0 & 0 & 0 & 1 \end{array}]$

$R_{3} = \frac{14}{215} R_{4} + R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = \frac{14}{215} R_{4} + R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ (\frac{14}{215}) \cdot (0) + 0 & (\frac{14}{215}) \cdot (0) + 0 & (\frac{14}{215}) \cdot (0) + 1 & (\frac{14}{215}) \cdot (1) - \frac{14}{215} \\ 0 & 0 & 0 & 1 \end{array}]$

$R_{3} = \frac{14}{215} R_{4} + R_{3}$

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

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

Find Reduced Row Echelon Form [[-2,0,4,0],[167,0,-119,-14],[15,70,-15,0],[-325,0,225,100]]

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