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

Problem

[[1,0,1],[3,0,3],[0,1,−1],[3,1,2]]

Solution

1. Identify the first pivot in the first column, which is the 1 at (R_1)

2. Eliminate the entries below the first pivot by performing row operations (R_2)−3*(R_1)→(R_2) and (R_4)−3*(R_1)→(R_4)

[[1,0,1],[0,0,0],[0,1,−1],[0,1,−1]]

3. Swap (R_2) and (R_3) to move a non-zero row up and establish the second pivot in the second column.

[[1,0,1],[0,1,−1],[0,0,0],[0,1,−1]]

4. Eliminate the entry below the second pivot by performing the row operation (R_4)−(R_2)→(R_4)

[[1,0,1],[0,1,−1],[0,0,0],[0,0,0]]

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

Final Answer

rref*([1,0,1],[3,0,3],[0,1,−1],[3,1,2])=[[1,0,1],[0,1,−1],[0,0,0],[0,0,0]]

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