Find Reduced Row Echelon Form [[1,-4,9,-7],[-1,2,-4,1],[5,-6,10,7]]

Problem

[[1,−4,9,−7],[−1,2,−4,1],[5,−6,10,7]]

Solution

1. Eliminate the first entry in the second and third rows by using the first row as a pivot. Add row 1 to row 2 ((R_2)+(R_1)→(R_2) and subtract 5 times row 1 from row 3 ((R_3)−5*(R_1)→(R_3).

[[1,−4,9,−7],[0,−2,5,−6],[0,14,−35,42]]

2. Normalize the second row by dividing it by −2 (−1/2*(R_2)→(R_2) to create a leading 1.

[[1,−4,9,−7],[0,1,−2.5,3],[0,14,−35,42]]

3. Eliminate the entries above and below the new leading 1 in the second column. Add 4 times row 2 to row 1 ((R_1)+4*(R_2)→(R_1) and subtract 14 times row 2 from row 3 ((R_3)−14*(R_2)→(R_3).

[[1,0,−1,5],[0,1,−2.5,3],[0,0,0,0]]

Final Answer

rref*[[1,−4,9,−7],[−1,2,−4,1],[5,−6,10,7]]=[[1,0,−1,5],[0,1,−2.5,3],[0,0,0,0]]

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