Simplify the Matrix

Problem

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

Solution

1. Identify the goal to simplify the matrix into its Reduced Row Echelon Form (RREF) using Gaussian elimination.

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

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

3. Eliminate the second entry in the third row by performing the row operation (R_3)←(R_3)−3*(R_2)

[[1,−1,1,0,0],[0,1,−3,−2,2],[0,0,10,6,−4]]

4. Normalize the third row by dividing by 10 ((R_3)←1/10*(R_3).

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

5. Eliminate the third entries in the first and second rows by performing (R_1)←(R_1)−(R_3) and (R_2)←(R_2)+3*(R_3)

[[1,−1,0,−3/5,2/5],[0,1,0,−1/5,4/5],[0,0,1,3/5,−2/5]]

6. Eliminate the second entry in the first row by performing (R_1)←(R_1)+(R_2)

[[1,0,0,−4/5,6/5],[0,1,0,−1/5,4/5],[0,0,1,3/5,−2/5]]

Final Answer

[[1,−1,1,0,0],[2,−1,−1,−2,2],[2,1,3,0,2]]=[[1,0,0,−4/5,6/5],[0,1,0,−1/5,4/5],[0,0,1,3/5,−2/5]]

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