Find Reduced Row Echelon Form [[2,0,6],[-3,5,-19],[4,1,10],[1,-2,7]]

Problem

[[2,0,6],[−3,5,−19],[4,1,10],[1,−2,7]]

Solution

1. Swap the first and fourth rows to get a 1 in the pivot position of the first row.

[[1,−2,7],[−3,5,−19],[4,1,10],[2,0,6]]

2. Eliminate the entries below the first pivot by performing (R_2)+3*(R_1)→(R_2) (R_3)−4*(R_1)→(R_3) and (R_4)−2*(R_1)→(R_4)

[[1,−2,7],[0,−1,2],[0,9,−18],[0,4,−8]]

3. Normalize the second row by multiplying by −1 to create a leading 1

[[1,−2,7],[0,1,−2],[0,9,−18],[0,4,−8]]

4. Eliminate the entries above and below the second pivot by performing (R_1)+2*(R_2)→(R_1) (R_3)−9*(R_2)→(R_3) and (R_4)−4*(R_2)→(R_4)

[[1,0,3],[0,1,−2],[0,0,0],[0,0,0]]

Final Answer

rref*[[2,0,6],[−3,5,−19],[4,1,10],[1,−2,7]]=[[1,0,3],[0,1,−2],[0,0,0],[0,0,0]]

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