Find Reduced Row Echelon Form [[1,-4,2],[-2,8,-9],[-1,7,0]]

Problem

[[1,−4,2],[−2,8,−9],[−1,7,0]]

Solution

1. Eliminate the first entries in the second and third rows by adding multiples of the first row.

(R_2)→(R_2)+2*(R_1)

[[1,−4,2],[0,0,−5],[−1,7,0]]

(R_3)→(R_3)+(R_1)

[[1,−4,2],[0,0,−5],[0,3,2]]

2. Swap the second and third rows to position a non-zero pivot in the second column.

(R_2)↔(R_3)

[[1,−4,2],[0,3,2],[0,0,−5]]

3. Normalize the second and third rows by dividing each row by its leading coefficient.

(R_2)→1/3*(R_2)

[[1,−4,2],[0,1,2/3],[0,0,−5]]

(R_3)→−1/5*(R_3)

[[1,−4,2],[0,1,2/3],[0,0,1]]

4. Eliminate the entries above the pivot in the third column.

(R_2)→(R_2)−2/3*(R_3)

[[1,−4,2],[0,1,0],[0,0,1]]

(R_1)→(R_1)−2*(R_3)

[[1,−4,0],[0,1,0],[0,0,1]]

5. Eliminate the entry above the pivot in the second column.

(R_1)→(R_1)+4*(R_2)

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

Final Answer

rref*([1,−4,2],[−2,8,−9],[−1,7,0])=[[1,0,0],[0,1,0],[0,0,1]]

Want more problems? Check here!