Find Reduced Row Echelon Form

Problem

[[1,0,0,0,0,1],[1,−1,1,−1,1,0.5],[1,1,1,1,1,2],[1,2,4,8,16,4],[1,3,9,27,81,8]]

Solution

1. Eliminate the first entry of rows 2, 3, 4, and 5 by subtracting row 1 from each.

(R_2)→(R_2)−(R_1)

(R_3)→(R_3)−(R_1)

(R_4)→(R_4)−(R_1)

(R_5)→(R_5)−(R_1)

[[1,0,0,0,0,1],[0,−1,1,−1,1,−0.5],[0,1,1,1,1,1],[0,2,4,8,16,3],[0,3,9,27,81,7]]

2. Normalize row 2 by multiplying by −1

(R_2)→−1⋅(R_2)

[[1,0,0,0,0,1],[0,1,−1,1,−1,0.5],[0,1,1,1,1,1],[0,2,4,8,16,3],[0,3,9,27,81,7]]

3. Eliminate the second entry of rows 3, 4, and 5 using row 2.

(R_3)→(R_3)−(R_2)

(R_4)→(R_4)−2*(R_2)

(R_5)→(R_5)−3*(R_2)

[[1,0,0,0,0,1],[0,1,−1,1,−1,0.5],[0,0,2,0,2,0.5],[0,0,6,6,18,2],[0,0,12,24,84,5.5]]

4. Normalize row 3 by dividing by 2

(R_3)→1/2*(R_3)

[[1,0,0,0,0,1],[0,1,−1,1,−1,0.5],[0,0,1,0,1,0.25],[0,0,6,6,18,2],[0,0,12,24,84,5.5]]

5. Eliminate the third entry of rows 2, 4, and 5 using row 3.

(R_2)→(R_2)+(R_3)

(R_4)→(R_4)−6*(R_3)

(R_5)→(R_5)−12*(R_3)

[[1,0,0,0,0,1],[0,1,0,1,0,0.75],[0,0,1,0,1,0.25],[0,0,0,6,12,0.5],[0,0,0,24,72,2.5]]

6. Normalize row 4 by dividing by 6

(R_4)→1/6*(R_4)

[[1,0,0,0,0,1],[0,1,0,1,0,0.75],[0,0,1,0,1,0.25],[0,0,0,1,2,1/12],[0,0,0,24,72,2.5]]

7. Eliminate the fourth entry of rows 2 and 5 using row 4.

(R_2)→(R_2)−(R_4)

(R_5)→(R_5)−24*(R_4)

[[1,0,0,0,0,1],[0,1,0,0,−2,2/3],[0,0,1,0,1,0.25],[0,0,0,1,2,1/12],[0,0,0,0,24,0.5]]

8. Normalize row 5 by dividing by 24

(R_5)→1/24*(R_5)

[[1,0,0,0,0,1],[0,1,0,0,−2,2/3],[0,0,1,0,1,0.25],[0,0,0,1,2,1/12],[0,0,0,0,1,1/48]]

9. Eliminate the fifth entry of rows 2, 3, and 4 using row 5.

(R_2)→(R_2)+2*(R_5)

(R_3)→(R_3)−(R_5)

(R_4)→(R_4)−2*(R_5)

[[1,0,0,0,0,1],[0,1,0,0,0,17/24],[0,0,1,0,0,11/48],[0,0,0,1,0,1/24],[0,0,0,0,1,1/48]]

Final Answer

RREF*[[1,0,0,0,0,1],[1,−1,1,−1,1,0.5],[1,1,1,1,1,2],[1,2,4,8,16,4],[1,3,9,27,81,8]]=[[1,0,0,0,0,1],[0,1,0,0,0,17/24],[0,0,1,0,0,11/48],[0,0,0,1,0,1/24],[0,0,0,0,1,1/48]]

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