Find Reduced Row Echelon Form

Problem

{[a−2*b+3*d=11],[4*a+b−d=4],[2*a−b+3*d=10])

Solution

1. Represent the system as an augmented matrix.

[[1,−2,3,11],[4,1,−1,4],[2,−1,3,10]]

2. Eliminate the first entries in the second and third rows by performing (R_2)−4*(R_1)⇒(R_2) and (R_3)−2*(R_1)⇒(R_3)

[[1,−2,3,11],[0,9,−13,−40],[0,3,−3,−12]]

3. Simplify the third row by dividing by 3 ((R_3)/3⇒(R_3) and then swap it with the second row ((R_2)↔(R_3) to create a simpler pivot.

[[1,−2,3,11],[0,1,−1,−4],[0,9,−13,−40]]

4. Eliminate the entry below the second pivot by performing (R_3)−9*(R_2)⇒(R_3)

[[1,−2,3,11],[0,1,−1,−4],[0,0,−4,−4]]

5. Normalize the third row by dividing by -4 ((R_3)/−4⇒(R_3).

[[1,−2,3,11],[0,1,−1,−4],[0,0,1,1]]

6. Eliminate the entries above the third pivot by performing (R_2)+(R_3)⇒(R_2) and (R_1)−3*(R_3)⇒(R_1)

[[1,−2,0,8],[0,1,0,−3],[0,0,1,1]]

7. Eliminate the entry above the second pivot by performing (R_1)+2*(R_2)⇒(R_1)

[[1,0,0,2],[0,1,0,−3],[0,0,1,1]]

Final Answer

RREF=[[1,0,0,2],[0,1,0,−3],[0,0,1,1]]

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