Find Reduced Row Echelon Form

Problem

[[3,−6,21],[0,42,−21],[2,24,0],[2,54,−15]]

Solution

1. Normalize the first row by dividing (R_1) by 3 to create a leading 1

1/3*(R_1)→(R_1)

[[1,−2,7],[0,42,−21],[2,24,0],[2,54,−15]]

2. Eliminate the first column entries below the leading 1 by performing (R_3)−2*(R_1)→(R_3) and (R_4)−2*(R_1)→(R_4)

(R_3)−2*(R_1)→(R_3)

(R_4)−2*(R_1)→(R_4)

[[1,−2,7],[0,42,−21],[0,28,−14],[0,58,−29]]

3. Normalize the second row by dividing (R_2) by 42 to create the next leading 1

1/42*(R_2)→(R_2)

[[1,−2,7],[0,1,−0.5],[0,28,−14],[0,58,−29]]

4. Eliminate the second column entries above and below the leading 1 in (R_2)

(R_1)+2*(R_2)→(R_1)

(R_3)−28*(R_2)→(R_3)

(R_4)−58*(R_2)→(R_4)

[[1,0,6],[0,1,−0.5],[0,0,0],[0,0,0]]

Final Answer

rref*[[3,−6,21],[0,42,−21],[2,24,0],[2,54,−15]]=[[1,0,6],[0,1,−0.5],[0,0,0],[0,0,0]]

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