Simplify the Matrix

Problem

[[3,6,−4,16],[2,−1,−9,−18],[−3,−7,8,−27]]

Solution

1. Eliminate the first entry of the third row by adding the first row to the third row ((R_3)←(R_3)+(R_1).

[[3,6,−4,16],[2,−1,−9,−18],[0,−1,4,−11]]

2. Eliminate the first entry of the second row by multiplying the second row by 3 and subtracting 2 times the first row ((R_2)←3*(R_2)−2*(R_1).

[[3,6,−4,16],[0,−15,−19,−86],[0,−1,4,−11]]

3. Swap the second and third rows to make calculations easier ((R_2)↔(R_3).

[[3,6,−4,16],[0,−1,4,−11],[0,−15,−19,−86]]

4. Eliminate the second entry of the third row by subtracting 15 times the second row from the third row ((R_3)←(R_3)−15*(R_2).

[[3,6,−4,16],[0,−1,4,−11],[0,0,−79,79]]

5. Scale the third row by dividing by −79 ((R_3)←(R_3)/(−79).

[[3,6,−4,16],[0,−1,4,−11],[0,0,1,−1]]

6. Eliminate the third entries of the first and second rows using the third row ((R_2)←(R_2)−4*(R_3) and (R_1)←(R_1)+4*(R_3).

[[3,6,0,12],[0,−1,0,−7],[0,0,1,−1]]

7. Eliminate the second entry of the first row by adding 6 times the second row to the first row ((R_1)←(R_1)+6*(R_2).

[[3,0,0,−30],[0,−1,0,−7],[0,0,1,−1]]

8. Scale the first and second rows to obtain the reduced row echelon form ((R_1)←(R_1)/3 and (R_2)←−(R_2).

[[1,0,0,−10],[0,1,0,7],[0,0,1,−1]]

Final Answer

[[3,6,−4,16],[2,−1,−9,−18],[−3,−7,8,−27]]=[[1,0,0,−10],[0,1,0,7],[0,0,1,−1]]

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