Find Reduced Row Echelon Form

Problem

[[1,1,0],[1,0,1],[1,0,1],[2,1,0],[2,1,0]]

Solution

1. Identify the first pivot in the first column, which is already 1 at (R_1)

2. Eliminate the entries below the first pivot by performing row operations (R_2)−(R_1)→(R_2) (R_3)−(R_1)→(R_3) (R_4)−2*(R_1)→(R_4) and (R_5)−2*(R_1)→(R_5)

[[1,1,0],[0,−1,1],[0,−1,1],[0,−1,0],[0,−1,0]]

3. Normalize the second row by multiplying by −1 to create a pivot of 1 at (R_2)

[[1,1,0],[0,1,−1],[0,−1,1],[0,−1,0],[0,−1,0]]

4. Eliminate the entries in the second column around the new pivot using (R_1)−(R_2)→(R_1) (R_3)+(R_2)→(R_3) (R_4)+(R_2)→(R_4) and (R_5)+(R_2)→(R_5)

[[1,0,1],[0,1,−1],[0,0,0],[0,0,−1],[0,0,−1]]

5. Swap (R_3) and (R_4) to move the non-zero row up, then multiply the new (R_3) by −1 to create the third pivot.

[[1,0,1],[0,1,−1],[0,0,1],[0,0,0],[0,0,−1]]

6. Eliminate the entries in the third column around the third pivot using (R_1)−(R_3)→(R_1) (R_2)+(R_3)→(R_2) and (R_5)+(R_3)→(R_5)

[[1,0,0],[0,1,0],[0,0,1],[0,0,0],[0,0,0]]

Final Answer

rref*[[1,1,0],[1,0,1],[1,0,1],[2,1,0],[2,1,0]]=[[1,0,0],[0,1,0],[0,0,1],[0,0,0],[0,0,0]]

Want more problems? Check here!