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

Problem

[[2,−3,−2],[2,1,1],[3,2,1]]

Solution

1. Create a leading one in the first row by subtracting the first row from the third row to get a 1 then swapping it to the top.

(R_3)−(R_1)→(R_3)

[[2,−3,−2],[2,1,1],[1,5,3]]

(R_1)↔(R_3)

[[1,5,3],[2,1,1],[2,−3,−2]]

2. Eliminate entries below the leading one in the first column.

(R_2)−2*(R_1)→(R_2)

[[1,5,3],[0,−9,−5],[2,−3,−2]]

(R_3)−2*(R_1)→(R_3)

[[1,5,3],[0,−9,−5],[0,−13,−8]]

3. Create a leading one in the second row by multiplying the second row by −1/9

−1/9*(R_2)→(R_2)

[[1,5,3],[0,1,5/9],[0,−13,−8]]

4. Eliminate entries above and below the leading one in the second column.

(R_1)−5*(R_2)→(R_1)

[[1,0,2/9],[0,1,5/9],[0,−13,−8]]

(R_3)+13*(R_2)→(R_3)

[[1,0,2/9],[0,1,5/9],[0,0,−7/9]]

5. Create a leading one in the third row by multiplying by the reciprocal.

−9/7*(R_3)→(R_3)

[[1,0,2/9],[0,1,5/9],[0,0,1]]

6. Eliminate entries above the leading one in the third column.

(R_1)−2/9*(R_3)→(R_1)

[[1,0,0],[0,1,5/9],[0,0,1]]

(R_2)−5/9*(R_3)→(R_2)

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

Final Answer

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

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