Find Reduced Row Echelon Form

Problem

[[2,4,−1,−2,2,6],[1,3,2,−7,3,9],[5,8,−7,6,1,4]]

Solution

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

(R_1)↔(R_2)

[[1,3,2,−7,3,9],[2,4,−1,−2,2,6],[5,8,−7,6,1,4]]

2. Eliminate the entries below the first pivot by subtracting multiples of the first row from the second and third rows.

(R_2)−2*(R_1)→(R_2)

(R_3)−5*(R_1)→(R_3)

[[1,3,2,−7,3,9],[0,−2,−5,12,−4,−12],[0,−7,−17,41,−14,−41]]

3. Normalize the second row by dividing by −2 to create a leading 1

−1/2*(R_2)→(R_2)

[[1,3,2,−7,3,9],[0,1,5/2,−6,2,6],[0,−7,−17,41,−14,−41]]

4. Eliminate the entry below the second pivot.

(R_3)+7*(R_2)→(R_3)

[[1,3,2,−7,3,9],[0,1,5/2,−6,2,6],[0,0,1/2,−1,0,1]]

5. Normalize the third row by multiplying by 2 to create a leading 1

2*(R_3)→(R_3)

[[1,3,2,−7,3,9],[0,1,5/2,−6,2,6],[0,0,1,−2,0,2]]

6. Eliminate the entries above the third pivot to begin the back-substitution process.

(R_2)−5/2*(R_3)→(R_2)

(R_1)−2*(R_3)→(R_1)

[[1,3,0,−3,3,5],[0,1,0,−1,2,1],[0,0,1,−2,0,2]]

7. Eliminate the entry above the second pivot to reach the final reduced form.

(R_1)−3*(R_2)→(R_1)

[[1,0,0,0,−3,2],[0,1,0,−1,2,1],[0,0,1,−2,0,2]]

Final Answer

rref*[[2,4,−1,−2,2,6],[1,3,2,−7,3,9],[5,8,−7,6,1,4]]=[[1,0,0,0,−3,2],[0,1,0,−1,2,1],[0,0,1,−2,0,2]]

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