Find Reduced Row Echelon Form [[1,-1,1,-1],[2,1,-3,8],[1,-2,3,-5]]

Problem

[[1,−1,1,−1],[2,1,−3,8],[1,−2,3,−5]]

Solution

1. Eliminate the first entries in the second and third rows by performing row operations (R_2)−2*(R_1)→(R_2) and (R_3)−(R_1)→(R_3)

[[1,−1,1,−1],[0,3,−5,10],[0,−1,2,−4]]

2. Swap the second and third rows to make the calculations easier, then multiply the new second row by −1 to get a leading 1

[[1,−1,1,−1],[0,−1,2,−4],[0,3,−5,10]]

[[1,−1,1,−1],[0,1,−2,4],[0,3,−5,10]]

3. Eliminate the entries above and below the leading 1 in the second column using (R_1)+(R_2)→(R_1) and (R_3)−3*(R_2)→(R_3)

[[1,0,−1,3],[0,1,−2,4],[0,0,1,−2]]

4. Eliminate the entries above the leading 1 in the third column using (R_1)+(R_3)→(R_1) and (R_2)+2*(R_3)→(R_2)

[[1,0,0,1],[0,1,0,0],[0,0,1,−2]]

Final Answer

[[1,−1,1,−1],[2,1,−3,8],[1,−2,3,−5]]=[[1,0,0,1],[0,1,0,0],[0,0,1,−2]]

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