Simplify the Matrix

Problem

[[3,5,2,0],[4,7,5,0],[1,1,−4,0],[2,9,6,0]]

Solution

1. Swap rows to place a 1 in the top-left position for easier calculations. Swap (R_1) and (R_3)

[[1,1,−4,0],[4,7,5,0],[3,5,2,0],[2,9,6,0]]

2. Eliminate the first column entries below the pivot. Perform (R_2)−4*(R_1)⇒(R_2) (R_3)−3*(R_1)⇒(R_3) and (R_4)−2*(R_1)⇒(R_4)

[[1,1,−4,0],[0,3,21,0],[0,2,14,0],[0,7,14,0]]

3. Normalize the second row by dividing (R_2) by 3.

[[1,1,−4,0],[0,1,7,0],[0,2,14,0],[0,7,14,0]]

4. Eliminate the second column entries below the pivot. Perform (R_3)−2*(R_2)⇒(R_3) and (R_4)−7*(R_2)⇒(R_4)

[[1,1,−4,0],[0,1,7,0],[0,0,0,0],[0,0,−35,0]]

5. Swap rows to move the zero row to the bottom. Swap (R_3) and (R_4)

[[1,1,−4,0],[0,1,7,0],[0,0,−35,0],[0,0,0,0]]

6. Normalize the third row by dividing (R_3) by -35.

[[1,1,−4,0],[0,1,7,0],[0,0,1,0],[0,0,0,0]]

7. Eliminate the third column entries above the pivot. Perform (R_2)−7*(R_3)⇒(R_2) and (R_1)+4*(R_3)⇒(R_1)

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

8. Eliminate the second column entry above the pivot. Perform (R_1)−(R_2)⇒(R_1)

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

Final Answer

[[3,5,2,0],[4,7,5,0],[1,1,−4,0],[2,9,6,0]]=[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,0]]

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