Find Reduced Row Echelon Form

Problem

[[1,1,1,1],[0.04,0.05,0.08,0.02],[0.04,−0.1,0,0]]

Solution

1. Eliminate the first entry of the second row by performing the row operation (R_2)−0.04*(R_1)→(R_2)

[[1,1,1,1],[0,0.01,0.04,−0.02],[0.04,−0.1,0,0]]

2. Eliminate the first entry of the third row by performing the row operation (R_3)−0.04*(R_1)→(R_3)

[[1,1,1,1],[0,0.01,0.04,−0.02],[0,−0.14,−0.04,−0.04]]

3. Normalize the second row by multiplying it by 100 (100*(R_2)→(R_2) to create a leading 1

[[1,1,1,1],[0,1,4,−2],[0,−0.14,−0.04,−0.04]]

4. Eliminate the second entry of the first row by performing (R_1)−(R_2)→(R_1)

[[1,0,−3,3],[0,1,4,−2],[0,−0.14,−0.04,−0.04]]

5. Eliminate the second entry of the third row by performing (R_3)+0.14*(R_2)→(R_3)

[[1,0,−3,3],[0,1,4,−2],[0,0,0.52,−0.32]]

6. Normalize the third row by dividing by 0.52 (1/0.52*(R_3)→(R_3). Note that 0.32/0.52=32/52=8/13

[[1,0,−3,3],[0,1,4,−2],[0,0,1,−8/13]]

7. Eliminate the third entry of the first row by performing (R_1)+3*(R_3)→(R_1)

[[1,0,0,3+3*(−8/13)],[0,1,4,−2],[0,0,1,−8/13]]

3−24/13=(39−24)/13=15/13

[[1,0,0,15/13],[0,1,4,−2],[0,0,1,−8/13]]

8. Eliminate the third entry of the second row by performing (R_2)−4*(R_3)→(R_2)

−2−4*(−8/13)=−2+32/13=(−26+32)/13=6/13

[[1,0,0,15/13],[0,1,0,6/13],[0,0,1,−8/13]]

Final Answer

rref*[[1,1,1,1],[0.04,0.05,0.08,0.02],[0.04,−0.1,0,0]]=[[1,0,0,15/13],[0,1,0,6/13],[0,0,1,−8/13]]

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