Find Reduced Row Echelon Form [[4,20,31],[6,-5,-6],[2,-11,-16]]

Problem

[[4,20,31],[6,−5,−6],[2,−11,−16]]

Solution

1. Swap rows to place a smaller leading coefficient in the first row. Swap (R_1) and (R_3)

[[2,−11,−16],[6,−5,−6],[4,20,31]]

2. Create zeros in the first column below the first pivot. Perform (R_2)→(R_2)−3*(R_1) and (R_3)→(R_3)−2*(R_1)

[[2,−11,−16],[0,28,42],[0,42,63]]

3. Simplify rows by dividing by common factors. Divide (R_2) by 14 and (R_3) by 21

[[2,−11,−16],[0,2,3],[0,2,3]]

4. Create a zero in the third row. Perform (R_3)→(R_3)−(R_2)

[[2,−11,−16],[0,2,3],[0,0,0]]

5. Eliminate the term above the second pivot. Perform (R_1)→(R_1)+5.5*(R_2)

[[2,0,0.5],[0,2,3],[0,0,0]]

6. Normalize the pivots to 1 by dividing (R_1) by 2 and (R_2) by 2

[[1,0,0.25],[0,1,1.5],[0,0,0]]

Final Answer

rref*[[4,20,31],[6,−5,−6],[2,−11,−16]]=[[1,0,1/4],[0,1,3/2],[0,0,0]]

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