Find Reduced Row Echelon Form [[3,5,31,-4],[-1,1,3,-4],[6,0,12,12]]=0

Problem

[[3,5,31,−4],[−1,1,3,−4],[6,0,12,12]]*[[x],[y],[z],[w]]=[[0],[0],[0]]

Solution

1. Write the augmented matrix for the system of equations.

[[3,5,31,−4,0],[−1,1,3,−4,0],[6,0,12,12,0]]

2. Swap rows to place a −1 in the first pivot position to make calculations easier.

(R_1)↔(R_2)⇒[[−1,1,3,−4,0],[3,5,31,−4,0],[6,0,12,12,0]]

3. Normalize the first row by multiplying by −1

−1*(R_1)→(R_1)⇒[[1,−1,−3,4,0],[3,5,31,−4,0],[6,0,12,12,0]]

4. Eliminate the first column entries below the pivot.

−3*(R_1)+(R_2)→(R_2)⇒[[1,−1,−3,4,0],[0,8,40,−16,0],[6,0,12,12,0]]

−6*(R_1)+(R_3)→(R_3)⇒[[1,−1,−3,4,0],[0,8,40,−16,0],[0,6,30,−12,0]]

5. Normalize the second row by dividing by 8

1/8*(R_2)→(R_2)⇒[[1,−1,−3,4,0],[0,1,5,−2,0],[0,6,30,−12,0]]

6. Eliminate the second column entries above and below the pivot.

(R_2)+(R_1)→(R_1)⇒[[1,0,2,2,0],[0,1,5,−2,0],[0,6,30,−12,0]]

−6*(R_2)+(R_3)→(R_3)⇒[[1,0,2,2,0],[0,1,5,−2,0],[0,0,0,0,0]]

Final Answer

rref*[[3,5,31,−4,0],[−1,1,3,−4,0],[6,0,12,12,0]]=[[1,0,2,2,0],[0,1,5,−2,0],[0,0,0,0,0]]

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