Simplify the Matrix [[0,3,-1,1],[1,5,-1,-4],[-3,6,2,11]]

Problem

[[0,3,−1,1],[1,5,−1,−4],[−3,6,2,11]]

Solution

1. Swap rows to place a leading 1 in the first row by moving the second row to the top.

[[1,5,−1,−4],[0,3,−1,1],[−3,6,2,11]]

2. Eliminate the entry in the first column of the third row by adding 3 times the first row to the third row ((R_3)←(R_3)+3*(R_1).

[[1,5,−1,−4],[0,3,−1,1],[0,21,−1,−1]]

3. Normalize the second row by dividing it by 3 to create a leading 1 ((R_2)←1/3*(R_2).

[[1,5,−1,−4],[0,1,−1/3,1/3],[0,21,−1,−1]]

4. Eliminate the entry in the second column of the third row by subtracting 21 times the second row from the third row ((R_3)←(R_3)−21*(R_2).

[[1,5,−1,−4],[0,1,−1/3,1/3],[0,0,6,−8]]

5. Normalize the third row by dividing it by 6 ((R_3)←1/6*(R_3).

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

6. Eliminate the entries above the leading 1 in the third column by adding 1/3*(R_3) to (R_2) and adding (R_3) to (R_1)

[[1,5,0,−16/3],[0,1,0,−1/9],[0,0,1,−4/3]]

7. Eliminate the entry above the leading 1 in the second column by subtracting 5 times the second row from the first row ((R_1)←(R_1)−5*(R_2).

[[1,0,0,−43/9],[0,1,0,−1/9],[0,0,1,−4/3]]

Final Answer

[[0,3,−1,1],[1,5,−1,−4],[−3,6,2,11]]=[[1,0,0,−43/9],[0,1,0,−1/9],[0,0,1,−4/3]]

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