Find Reduced Row Echelon Form [[4,-3,1,0],[1,0,-2,0],[-2,1,1,0]]

Problem

[[4,−3,1,0],[1,0,−2,0],[−2,1,1,0]]

Solution

1. Swap rows to place a 1 in the first pivot position by moving the second row to the top.

(R_1)↔(R_2)

[[1,0,−2,0],[4,−3,1,0],[−2,1,1,0]]

2. Eliminate the entries below the first pivot by subtracting 4 times the first row from the second row and adding 2 times the first row to the third row.

(R_2)−4*(R_1)→(R_2)

(R_3)+2*(R_1)→(R_3)

[[1,0,−2,0],[0,−3,9,0],[0,1,−3,0]]

3. Normalize the second row by dividing it by −3 to create the second pivot.

−1/3*(R_2)→(R_2)

[[1,0,−2,0],[0,1,−3,0],[0,1,−3,0]]

4. Eliminate the entry below the second pivot by subtracting the second row from the third row.

(R_3)−(R_2)→(R_3)

[[1,0,−2,0],[0,1,−3,0],[0,0,0,0]]

Final Answer

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

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