Find Reduced Row Echelon Form [[1/2,-1,-1/3],[3/4,1/2,-1],[1,-4,1]]

Problem

[[1/2,−1,−1/3],[3/4,1/2,−1],[1,−4,1]]

Solution

1. Multiply the first row (R_1) by 2 to create a leading 1 in the first column.

[[1,−2,−2/3],[3/4,1/2,−1],[1,−4,1]]

2. Eliminate the first column entries below the leading 1 by performing (R_2)−3/4*(R_1)⇒(R_2) and (R_3)−(R_1)⇒(R_3)

[[1,−2,−2/3],[0,2,−1/2],[0,−2,5/3]]

3. Normalize the second row by multiplying (R_2) by 1/2 to create a leading 1

[[1,−2,−2/3],[0,1,−1/4],[0,−2,5/3]]

4. Eliminate the second column entries above and below the leading 1 by performing (R_1)+2*(R_2)⇒(R_1) and (R_3)+2*(R_2)⇒(R_3)

[[1,0,−7/6],[0,1,−1/4],[0,0,7/6]]

5. Normalize the third row by multiplying (R_3) by 6/7 to create a leading 1

[[1,0,−7/6],[0,1,−1/4],[0,0,1]]

6. Eliminate the third column entries above the leading 1 by performing (R_1)+7/6*(R_3)⇒(R_1) and (R_2)+1/4*(R_3)⇒(R_2)

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

Final Answer

rref*[[1/2,−1,−1/3],[3/4,1/2,−1],[1,−4,1]]=[[1,0,0],[0,1,0],[0,0,1]]

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