Find Reduced Row Echelon Form [[1,2,3],[1,1,2],[2,0,2]]

Problem

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

Solution

1. Identify the first pivot in the first row, which is 1 at position (1,1)

2. Eliminate the entries below the first pivot by performing row operations (R_2)→(R_2)−(R_1) and (R_3)→(R_3)−2*(R_1)

[[1,2,3],[0,−1,−1],[0,−4,−4]]

3. Normalize the second row by multiplying (R_2) by −1 to create a pivot of 1 at position (2,2)

[[1,2,3],[0,1,1],[0,−4,−4]]

4. Eliminate the entries above and below the second pivot by performing (R_1)→(R_1)−2*(R_2) and (R_3)→(R_3)+4*(R_2)

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

5. Verify that the matrix is in reduced row echelon form, as all pivots are 1 they are the only non-zero entries in their columns, and any zero rows are at the bottom.

Final Answer

rref*[[1,2,3],[1,1,2],[2,0,2]]=[[1,0,1],[0,1,1],[0,0,0]]

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