Find Reduced Row Echelon Form

Problem

rref*[[1,1,1,0,5],[0,1,0,0,5],[1,0,0,1,6],[0,0,1,0,6]]

Solution

1. Identify the initial matrix A

A=[[1,1,1,0,5],[0,1,0,0,5],[1,0,0,1,6],[0,0,1,0,6]]

2. Eliminate the first entry in the third row by performing the row operation (R_3)←(R_3)−(R_1)

[[1,1,1,0,5],[0,1,0,0,5],[0,−1,−1,1,1],[0,0,1,0,6]]

3. Eliminate the second entry in the third row by performing (R_3)←(R_3)+(R_2)

[[1,1,1,0,5],[0,1,0,0,5],[0,0,−1,1,6],[0,0,1,0,6]]

4. Eliminate the third entry in the fourth row by performing (R_4)←(R_4)+(R_3)

[[1,1,1,0,5],[0,1,0,0,5],[0,0,−1,1,6],[0,0,0,1,12]]

5. Normalize the third row by multiplying by −1 ((R_3)←−1⋅(R_3).

[[1,1,1,0,5],[0,1,0,0,5],[0,0,1,−1,−6],[0,0,0,1,12]]

6. Eliminate entries above the pivot in the fourth column by performing (R_3)←(R_3)+(R_4)

[[1,1,1,0,5],[0,1,0,0,5],[0,0,1,0,6],[0,0,0,1,12]]

7. Eliminate entries above the pivot in the third column by performing (R_1)←(R_1)−(R_3)

[[1,1,0,0,−1],[0,1,0,0,5],[0,0,1,0,6],[0,0,0,1,12]]

8. Eliminate entries above the pivot in the second column by performing (R_1)←(R_1)−(R_2)

[[1,0,0,0,−6],[0,1,0,0,5],[0,0,1,0,6],[0,0,0,1,12]]

Final Answer

rref*[[1,1,1,0,5],[0,1,0,0,5],[1,0,0,1,6],[0,0,1,0,6]]=[[1,0,0,0,−6],[0,1,0,0,5],[0,0,1,0,6],[0,0,0,1,12]]

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