Find Reduced Row Echelon Form

Problem

[[1,2,1,3,10],[−1,1,3,−4,−15],[8,7,−4,36,131]]

Solution

1. Eliminate the first entry in the second and third rows by adding the first row to the second row ((R_2)+(R_1)→(R_2) and subtracting 8 times the first row from the third row ((R_3)−8*(R_1)→(R_3).

[[1,2,1,3,10],[0,3,4,−1,−5],[0,−9,−12,12,51]]

2. Eliminate the second entry in the third row by adding 3 times the second row to the third row ((R_3)+3*(R_2)→(R_3).

[[1,2,1,3,10],[0,3,4,−1,−5],[0,0,0,9,36]]

3. Normalize the third row by dividing it by 9 ((R_3)/9→(R_3).

[[1,2,1,3,10],[0,3,4,−1,−5],[0,0,0,1,4]]

4. Eliminate the fourth entry in the first and second rows by subtracting 3 times the third row from the first row ((R_1)−3*(R_3)→(R_1) and adding the third row to the second row ((R_2)+(R_3)→(R_2).

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

5. Normalize the second row by dividing it by 3 ((R_2)/3→(R_2).

[[1,2,1,0,−2],[0,1,4/3,0,−1/3],[0,0,0,1,4]]

6. Eliminate the second entry in the first row by subtracting 2 times the second row from the first row ((R_1)−2*(R_2)→(R_1).

[[1,0,−5/3,0,−4/3],[0,1,4/3,0,−1/3],[0,0,0,1,4]]

Final Answer

[[1,2,1,3,10],[−1,1,3,−4,−15],[8,7,−4,36,131]]=[[1,0,−5/3,0,−4/3],[0,1,4/3,0,−1/3],[0,0,0,1,4]]

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