Find Reduced Row Echelon Form

Problem

[[3,7,8,90],[2,−2,2,10],[4,7,8,92],[3,4,6,64]]

Solution

1. Create a leading one in the first row by subtracting the second row from the first row ((R_1)−(R_2)→(R_1).

[[1,9,6,80],[2,−2,2,10],[4,7,8,92],[3,4,6,64]]

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

[[1,9,6,80],[0,−20,−10,−150],[0,−29,−16,−228],[0,−23,−12,−176]]

3. Simplify the second row by dividing by −10 ((R_2)÷−10→(R_2).

[[1,9,6,80],[0,2,1,15],[0,−29,−16,−228],[0,−23,−12,−176]]

4. Eliminate entries in the second column by performing (R_3)+14*(R_2)→(R_3) and (R_4)+11*(R_2)→(R_4)

[[1,9,6,80],[0,2,1,15],[0,−1,−2,−18],[0,−1,−1,−11]]

5. Swap rows to get a simpler pivot ((R_2)↔(R_3) and then multiply the new (R_2) by −1

[[1,9,6,80],[0,1,2,18],[0,2,1,15],[0,−1,−1,−11]]

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

[[1,0,−12,−82],[0,1,2,18],[0,0,−3,−21],[0,0,1,7]]

7. Create a leading one in the third row by dividing by −3 ((R_3)÷−3→(R_3).

[[1,0,−12,−82],[0,1,2,18],[0,0,1,7],[0,0,1,7]]

8. Eliminate entries above and below the third pivot using (R_1)+12*(R_3)→(R_1) (R_2)−2*(R_3)→(R_2) and (R_4)−(R_3)→(R_4)

[[1,0,0,2],[0,1,0,4],[0,0,1,7],[0,0,0,0]]

Final Answer

rref*[[3,7,8,90],[2,−2,2,10],[4,7,8,92],[3,4,6,64]]=[[1,0,0,2],[0,1,0,4],[0,0,1,7],[0,0,0,0]]

Want more problems? Check here!