Find Reduced Row Echelon Form [[1,7,-2,-38],[2,13,2,-71],[1,4,-5,-23]]

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ 2 & 13 & 2 & - 71 \\ 1 & 4 & - 5 & - 23 \end{array}]$

Step 1

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

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} \\ 1 & 4 & - 5 & - 23 \end{array}]$

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

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

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ (- 2) \cdot (1) + 2 & (- 2) \cdot (7) + 13 & (- 2) \cdot (- 2) + 2 & (- 2) \cdot (- 38) - 71 \\ 1 & 4 & - 5 & - 23 \end{array}]$

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

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

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ 0 & - 1 & 6 & 5 \\ 1 & 4 & - 5 & - 23 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ 0 & - 1 & 6 & 5 \\ - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} \end{array}]$

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

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

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ 0 & - 1 & 6 & 5 \\ (- 1) \cdot (1) + 1 & (- 1) \cdot (7) + 4 & (- 1) \cdot (- 2) - 5 & (- 1) \cdot (- 38) - 23 \end{array}]$

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

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

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ 0 & - 1 & 6 & 5 \\ 0 & - 3 & - 3 & 15 \end{array}]$

Step 3

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 $1$ .

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ - 1 \cdot R_{2} & - 1 \cdot R_{2} & - 1 \cdot R_{2} & - 1 \cdot R_{2} \\ 0 & - 3 & - 3 & 15 \end{array}]$

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

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

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ (- 1) \cdot (0) & (- 1) \cdot (- 1) & (- 1) \cdot (6) & (- 1) \cdot (5) \\ 0 & - 3 & - 3 & 15 \end{array}]$

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

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

$[\begin{array}{c} 1 & 7 & - 2 & - 38 \\ 0 & 1 & - 6 & - 5 \\ 0 & - 3 & - 3 & 15 \end{array}]$

Step 4

Perform the row operation $R_{1} = - 7 \cdot R_{2} + R_{1}$ 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 $R_{1} = - 7 \cdot R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - 7 \cdot R_{2} + R_{1} & - 7 \cdot R_{2} + R_{1} & - 7 \cdot R_{2} + R_{1} & - 7 \cdot R_{2} + R_{1} \\ 0 & 1 & - 6 & - 5 \\ 0 & - 3 & - 3 & 15 \end{array}]$

$R_{1} = - 7 \cdot R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - 7 \cdot R_{2} + R_{1}$ .

$[\begin{array}{c} (- 7) \cdot (0) + 1 & (- 7) \cdot (1) + 7 & (- 7) \cdot (- 6) - 2 & (- 7) \cdot (- 5) - 38 \\ 0 & 1 & - 6 & - 5 \\ 0 & - 3 & - 3 & 15 \end{array}]$

$R_{1} = - 7 \cdot R_{2} + R_{1}$

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

$[\begin{array}{c} 1 & 0 & 40 & - 3 \\ 0 & 1 & - 6 & - 5 \\ 0 & - 3 & - 3 & 15 \end{array}]$

Step 5

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 $0$ .

$[\begin{array}{c} 1 & 0 & 40 & - 3 \\ 0 & 1 & - 6 & - 5 \\ 3 \cdot R_{2} + R_{3} & 3 \cdot R_{2} + R_{3} & 3 \cdot R_{2} + R_{3} & 3 \cdot R_{2} + R_{3} \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & 40 & - 3 \\ 0 & 1 & - 6 & - 5 \\ (3) \cdot (0) + 0 & (3) \cdot (1) - 3 & (3) \cdot (- 6) - 3 & (3) \cdot (- 5) + 15 \end{array}]$

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

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

$[\begin{array}{c} 1 & 0 & 40 & - 3 \\ 0 & 1 & - 6 & - 5 \\ 0 & 0 & - 21 & 0 \end{array}]$

Step 6

Perform the row operation $R_{3} = - \frac{1}{21} R_{3}$ 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 $R_{3} = - \frac{1}{21} R_{3}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & 40 & - 3 \\ 0 & 1 & - 6 & - 5 \\ - \frac{1}{21} R_{3} & - \frac{1}{21} R_{3} & - \frac{1}{21} R_{3} & - \frac{1}{21} R_{3} \end{array}]$

$R_{3} = - \frac{1}{21} R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - \frac{1}{21} R_{3}$ .

$[\begin{array}{c} 1 & 0 & 40 & - 3 \\ 0 & 1 & - 6 & - 5 \\ (- \frac{1}{21}) \cdot (0) & (- \frac{1}{21}) \cdot (0) & (- \frac{1}{21}) \cdot (- 21) & (- \frac{1}{21}) \cdot (0) \end{array}]$

$R_{3} = - \frac{1}{21} R_{3}$

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

$[\begin{array}{c} 1 & 0 & 40 & - 3 \\ 0 & 1 & - 6 & - 5 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 7

Perform the row operation $R_{1} = - 40 \cdot R_{3} + R_{1}$ 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 $R_{1} = - 40 \cdot R_{3} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - 40 \cdot R_{3} + R_{1} & - 40 \cdot R_{3} + R_{1} & - 40 \cdot R_{3} + R_{1} & - 40 \cdot R_{3} + R_{1} \\ 0 & 1 & - 6 & - 5 \\ 0 & 0 & 1 & 0 \end{array}]$

$R_{1} = - 40 \cdot R_{3} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - 40 \cdot R_{3} + R_{1}$ .

$[\begin{array}{c} (- 40) \cdot (0) + 1 & (- 40) \cdot (0) + 0 & (- 40) \cdot (1) + 40 & (- 40) \cdot (0) - 3 \\ 0 & 1 & - 6 & - 5 \\ 0 & 0 & 1 & 0 \end{array}]$

$R_{1} = - 40 \cdot R_{3} + R_{1}$

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

$[\begin{array}{c} 1 & 0 & 0 & - 3 \\ 0 & 1 & - 6 & - 5 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 8

Perform the row operation $R_{2} = 6 \cdot R_{3} + R_{2}$ 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 $R_{2} = 6 \cdot R_{3} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

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

$R_{2} = 6 \cdot R_{3} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = 6 \cdot R_{3} + R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 3 \\ (6) \cdot (0) + 0 & (6) \cdot (0) + 1 & (6) \cdot (1) - 6 & (6) \cdot (0) - 5 \\ 0 & 0 & 1 & 0 \end{array}]$

$R_{2} = 6 \cdot R_{3} + R_{2}$

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

$[\begin{array}{c} 1 & 0 & 0 & - 3 \\ 0 & 1 & 0 & - 5 \\ 0 & 0 & 1 & 0 \end{array}]$

Find Reduced Row Echelon Form [[1,7,-2,-38],[2,13,2,-71],[1,4,-5,-23]]

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