Find Reduced Row Echelon Form

Problem

[[1,1,1,3200],[0.07,0.08,0.09,274],[0,−1,1,1000]]

Solution

1. Eliminate the first entry in the second row by performing the row operation (R_2)−0.07*(R_1)→(R_2)

(R_2)−0.07*(1,1,1,3200)=(0.07−0.07,0.08−0.07,0.09−0.07,274−224)

[[1,1,1,3200],[0,0.01,0.02,50],[0,−1,1,1000]]

2. Normalize the second row by multiplying (R_2) by 100 to create a leading one.

100*(R_2)→(R_2)

[[1,1,1,3200],[0,1,2,5000],[0,−1,1,1000]]

3. Eliminate the entries in the second column for the first and third rows using (R_1)−(R_2)→(R_1) and (R_3)+(R_2)→(R_3)

(R_1)−(R_2)=(1−0,1−1,1−2,3200−5000)=(1,0,−1,−1800)

(R_3)+(R_2)=(0+0,−1+1,1+2,1000+5000)=(0,0,3,6000)

[[1,0,−1,−1800],[0,1,2,5000],[0,0,3,6000]]

4. Normalize the third row by dividing (R_3) by 3

1/3*(R_3)→(R_3)

[[1,0,−1,−1800],[0,1,2,5000],[0,0,1,2000]]

5. Eliminate the entries in the third column for the first and second rows using (R_1)+(R_3)→(R_1) and (R_2)−2*(R_3)→(R_2)

(R_1)+(R_3)=(1+0,0+0,−1+1,−1800+2000)=(1,0,0,200)

(R_2)−2*(R_3)=(0−0,1−0,2−2,5000−4000)=(0,1,0,1000)

[[1,0,0,200],[0,1,0,1000],[0,0,1,2000]]

Final Answer

rref*[[1,1,1,3200],[0.07,0.08,0.09,274],[0,−1,1,1000]]=[[1,0,0,200],[0,1,0,1000],[0,0,1,2000]]

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