Find Reduced Row Echelon Form [[0,1,5,-4],[1,4,3,-2],[2,7,1,-2]]

Problem

[[0,1,5,−4],[1,4,3,−2],[2,7,1,−2]]

Solution

1. Swap the first and second rows to place a leading 1 in the top-left corner.

(R_1)↔(R_2)

[[1,4,3,−2],[0,1,5,−4],[2,7,1,−2]]

2. Eliminate the 2 in the first column of the third row by subtracting twice the first row from the third row.

(R_3)−2*(R_1)→(R_3)

[[1,4,3,−2],[0,1,5,−4],[0,−1,−5,2]]

3. Eliminate the −1 in the second column of the third row by adding the second row to the third row.

(R_3)+(R_2)→(R_3)

[[1,4,3,−2],[0,1,5,−4],[0,0,0,−2]]

4. Normalize the third row by dividing it by −2 to create a leading 1

−1/2*(R_3)→(R_3)

[[1,4,3,−2],[0,1,5,−4],[0,0,0,1]]

5. Eliminate the entries above the leading 1 in the fourth column by using the third row.

(R_2)+4*(R_3)→(R_2)

[[1,4,3,−2],[0,1,5,0],[0,0,0,1]]

(R_1)+2*(R_3)→(R_1)

[[1,4,3,0],[0,1,5,0],[0,0,0,1]]

6. Eliminate the 4 in the second column of the first row by subtracting four times the second row from the first row.

(R_1)−4*(R_2)→(R_1)

[[1,0,−17,0],[0,1,5,0],[0,0,0,1]]

Final Answer

rref*[[0,1,5,−4],[1,4,3,−2],[2,7,1,−2]]=[[1,0,−17,0],[0,1,5,0],[0,0,0,1]]

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