Find Reduced Row Echelon Form

Problem

[[1,1,−2,0,−3,−1],[1,2,−3,0,−5,0],[1,3,−4,0,−13,−11],[1,−1,1,1,−10,0],[1,4,−5,0,−1,23]]

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)

[[1,1,−2,0,−3,−1],[0,1,−1,0,−2,1],[0,2,−2,0,−10,−10],[0,−2,3,1,−7,1],[0,3,−3,0,2,24]]

2. Eliminate the second entry of rows 1, 3, 4, and 5 using row 2.

(R_1)→(R_1)−(R_2)

(R_3)→(R_3)−2*(R_2)

(R_4)→(R_4)+2*(R_2)

(R_5)→(R_5)−3*(R_2)

[[1,0,−1,0,−1,−2],[0,1,−1,0,−2,1],[0,0,0,0,−6,−12],[0,0,1,1,−11,3],[0,0,0,0,8,21]]

3. Swap row 3 and row 4 to bring the next pivot into position.

(R_3)↔(R_4)

[[1,0,−1,0,−1,−2],[0,1,−1,0,−2,1],[0,0,1,1,−11,3],[0,0,0,0,−6,−12],[0,0,0,0,8,21]]

4. Eliminate the third entry of rows 1 and 2 using row 3.

(R_1)→(R_1)+(R_3)

(R_2)→(R_2)+(R_3)

[[1,0,0,1,−12,1],[0,1,0,1,−13,4],[0,0,1,1,−11,3],[0,0,0,0,−6,−12],[0,0,0,0,8,21]]

5. Normalize row 4 by dividing by -6.

(R_4)→−1/6*(R_4)

[[1,0,0,1,−12,1],[0,1,0,1,−13,4],[0,0,1,1,−11,3],[0,0,0,0,1,2],[0,0,0,0,8,21]]

6. Eliminate the fifth entry of rows 1, 2, 3, and 5 using row 4.

(R_1)→(R_1)+12*(R_4)

(R_2)→(R_2)+13*(R_4)

(R_3)→(R_3)+11*(R_4)

(R_5)→(R_5)−8*(R_4)

[[1,0,0,1,0,25],[0,1,0,1,0,30],[0,0,1,1,0,25],[0,0,0,0,1,2],[0,0,0,0,0,5]]

7. Normalize row 5 by dividing by 5.

(R_5)→1/5*(R_5)

[[1,0,0,1,0,25],[0,1,0,1,0,30],[0,0,1,1,0,25],[0,0,0,0,1,2],[0,0,0,0,0,1]]

8. Eliminate the sixth entry of rows 1, 2, 3, and 4 using row 5.

(R_1)→(R_1)−25*(R_5)

(R_2)→(R_2)−30*(R_5)

(R_3)→(R_3)−25*(R_5)

(R_4)→(R_4)−2*(R_5)

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

Final Answer

RREF*[[1,1,−2,0,−3,−1],[1,2,−3,0,−5,0],[1,3,−4,0,−13,−11],[1,−1,1,1,−10,0],[1,4,−5,0,−1,23]]=[[1,0,0,1,0,0],[0,1,0,1,0,0],[0,0,1,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]

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