Find Reduced Row Echelon Form

Problem

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

Solution

1. Reverse the row order to move the leading zeros to the right and establish a staircase pattern. Swap (R_1) with (R_4) and (R_2) with (R_3)

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

2. Eliminate the entry in (R_1) above the pivot in (R_4) by performing (R_1)→(R_1)−(R_4)

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

3. Eliminate the entry in (R_2) above the pivot in (R_4) by performing (R_2)→(R_2)−(R_4)

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

4. Eliminate the entry in (R_1) above the pivot in (R_3) by performing (R_1)→(R_1)−(R_3)

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

Final Answer

[[0,0,0,0,1,0,1,1],[0,0,0,1,0,1,1,0],[0,0,1,0,1,1,0,0],[0,1,0,1,1,0,0,0]]→[[0,1,0,0,0,−1,−2,−1],[0,0,1,0,0,1,−1,−1],[0,0,0,1,0,1,1,0],[0,0,0,0,1,0,1,1]]

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