Simplify the Matrix

Problem

[[1,−3,4,5,3],[0,1,5,0,12],[0,0,1,−1,2],[1,−3,0,0,−5]]

Solution

1. Identify the goal to simplify the matrix into its Reduced Row Echelon Form (RREF) using Gaussian elimination.

2. Eliminate the first entry in the fourth row by performing the row operation (R_4)→(R_4)−(R_1)

[[1,−3,4,5,3],[0,1,5,0,12],[0,0,1,−1,2],[0,0,−4,−5,−8]]

3. Eliminate the third entry in the fourth row by performing the row operation (R_4)→(R_4)+4*(R_3)

[[1,−3,4,5,3],[0,1,5,0,12],[0,0,1,−1,2],[0,0,0,−9,0]]

4. Normalize the fourth row by dividing by −9 using the operation (R_4)→−1/9*(R_4)

[[1,−3,4,5,3],[0,1,5,0,12],[0,0,1,−1,2],[0,0,0,1,0]]

5. Eliminate entries above the pivot in the fourth column using (R_3)→(R_3)+(R_4) and (R_1)→(R_1)−5*(R_4)

[[1,−3,4,0,3],[0,1,5,0,12],[0,0,1,0,2],[0,0,0,1,0]]

6. Eliminate entries above the pivot in the third column using (R_2)→(R_2)−5*(R_3) and (R_1)→(R_1)−4*(R_3)

[[1,−3,0,0,−5],[0,1,0,0,2],[0,0,1,0,2],[0,0,0,1,0]]

7. Eliminate the entry above the pivot in the second column using (R_1)→(R_1)+3*(R_2)

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

Final Answer

[[1,−3,4,5,3],[0,1,5,0,12],[0,0,1,−1,2],[1,−3,0,0,−5]]=[[1,0,0,0,1],[0,1,0,0,2],[0,0,1,0,2],[0,0,0,1,0]]

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