Find Reduced Row Echelon Form [[3,-2,0],[8,-3,4],[13,-2,0]]

Problem

[[3,−2,0],[8,−3,4],[13,−2,0]]

Solution

1. Divide the first row (R_1) by 3 to create a leading 1 in the first column.

[[1,−2/3,0],[8,−3,4],[13,−2,0]]

2. Eliminate the first column entries below the leading 1 by performing (R_2)−8*(R_1)→(R_2) and (R_3)−13*(R_1)→(R_3)

[[1,−2/3,0],[0,7/3,4],[0,20/3,0]]

3. Divide the third row (R_3) by 20/3 to create a leading 1 in the second column (swapping or scaling for simplicity).

[[1,−2/3,0],[0,7/3,4],[0,1,0]]

4. Swap (R_2) and (R_3) to position the leading 1 correctly.

[[1,−2/3,0],[0,1,0],[0,7/3,4]]

5. Eliminate the second column entries above and below the leading 1 in (R_2) by performing (R_1)+2/3*(R_2)→(R_1) and (R_3)−7/3*(R_2)→(R_3)

[[1,0,0],[0,1,0],[0,0,4]]

6. Divide the third row (R_3) by 4 to create a leading 1 in the third column.

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

Final Answer

rref*[[3,−2,0],[8,−3,4],[13,−2,0]]=[[1,0,0],[0,1,0],[0,0,1]]

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