Find Reduced Row Echelon Form [[1,0,-3,8],[2,2,9,7],[0,1,5,-2]]

Problem

[[1,0,−3,8],[2,2,9,7],[0,1,5,−2]]

Solution

1. Identify the first pivot in the first row, which is already 1 in the first column.

2. Eliminate the entry below the first pivot by performing the row operation (R_2)−2*(R_1)⇒(R_2)

[[1,0,−3,8],[0,2,15,−9],[0,1,5,−2]]

3. Swap the second and third rows ((R_2)⇔(R_3) to obtain a 1 in the second pivot position.

[[1,0,−3,8],[0,1,5,−2],[0,2,15,−9]]

4. Eliminate the entry below the second pivot by performing the row operation (R_3)−2*(R_2)⇒(R_3)

[[1,0,−3,8],[0,1,5,−2],[0,0,5,−5]]

5. Normalize the third row by dividing by 5 (1/5*(R_3)⇒(R_3) to create the third pivot.

[[1,0,−3,8],[0,1,5,−2],[0,0,1,−1]]

6. Eliminate the entries above the third pivot by performing (R_2)−5*(R_3)⇒(R_2) and (R_1)+3*(R_3)⇒(R_1)

[[1,0,0,5],[0,1,0,3],[0,0,1,−1]]

Final Answer

[[1,0,−3,8],[2,2,9,7],[0,1,5,−2]]=[[1,0,0,5],[0,1,0,3],[0,0,1,−1]]

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