Find Reduced Row Echelon Form

Problem

[[1,0.5,0.33,0.25,1.1],[0.5,0.33,0.25,0.21,1.2],[0.33,0.25,0.2,0.17,1.3],[0.25,0.2,0.17,0.14,1.4]]

Solution

1. Eliminate the first column entries below the first pivot by performing row operations (R_2)→(R_2)−0.5*(R_1) (R_3)→(R_3)−0.33*(R_1) and (R_4)→(R_4)−0.25*(R_1)

[[1,0.5,0.33,0.25,1.1],[0,0.08,0.085,0.085,0.65],[0,0.085,0.0911,0.0875,0.937],[0,0.075,0.0875,0.0775,1.125]]

2. Normalize the second row by dividing (R_2) by 0.08 to create a pivot of 1 in the second column.

[[1,0.5,0.33,0.25,1.1],[0,1,1.0625,1.0625,8.125],[0,0.085,0.0911,0.0875,0.937],[0,0.075,0.0875,0.0775,1.125]]

3. Eliminate the second column entries above and below the second pivot using (R_1)→(R_1)−0.5*(R_2) (R_3)→(R_3)−0.085*(R_2) and (R_4)→(R_4)−0.075*(R_2)

[[1,0,−0.20125,−0.28125,−2.9625],[0,1,1.0625,1.0625,8.125],[0,0,0.0007875,−0.0028125,0.246375],[0,0,0.0078125,−0.0021875,0.515625]]

4. Normalize the third row by dividing (R_3) by 0.0007875 to create a pivot of 1

[[1,0,−0.20125,−0.28125,−2.9625],[0,1,1.0625,1.0625,8.125],[0,0,1,−3.5714,312.857],[0,0,0.0078125,−0.0021875,0.515625]]

5. Eliminate the third column entries above and below the third pivot using (R_1)→(R_1)+0.20125*(R_3) (R_2)→(R_2)−1.0625*(R_3) and (R_4)→(R_4)−0.0078125*(R_3)

[[1,0,0,−1,60],[0,1,0,4.857,−324.286],[0,0,1,−3.5714,312.857],[0,0,0,0.0257,−1.9286]]

6. Normalize the fourth row by dividing (R_4) by 0.0257 to create a pivot of 1

[[1,0,0,−1,60],[0,1,0,4.857,−324.286],[0,0,1,−3.5714,312.857],[0,0,0,1,−75]]

7. Eliminate the fourth column entries above the fourth pivot using (R_1)→(R_1)+(R_4) (R_2)→(R_2)−4.857*(R_4) and (R_3)→(R_3)+3.5714*(R_4)

[[1,0,0,0,−15],[0,1,0,0,40],[0,0,1,0,45],[0,0,0,1,−75]]

Final Answer

rref*([1,0.5,0.33,0.25,1.1],[0.5,0.33,0.25,0.21,1.2],[0.33,0.25,0.2,0.17,1.3],[0.25,0.2,0.17,0.14,1.4])=[[1,0,0,0,−15],[0,1,0,0,40],[0,0,1,0,45],[0,0,0,1,−75]]

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