Find Reduced Row Echelon Form

Problem

[[1,0,1,−1],[2,−1,0,2],[0,1,2,−4],[a,b,c,3]]

Solution

1. Eliminate the first entry in the second row by performing (R_2)−2*(R_1)→(R_2)

[[1,0,1,−1],[0,−1,−2,4],[0,1,2,−4],[a,b,c,3]]

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

[[1,0,1,−1],[0,−1,−2,4],[0,1,2,−4],[0,b,c−a,3+a]]

3. Normalize the second row by multiplying by −1 performing −1*(R_2)→(R_2)

[[1,0,1,−1],[0,1,2,−4],[0,1,2,−4],[0,b,c−a,3+a]]

4. Eliminate the second entry in the third row by performing (R_3)−(R_2)→(R_3)

[[1,0,1,−1],[0,1,2,−4],[0,0,0,0],[0,b,c−a,3+a]]

5. Eliminate the second entry in the fourth row by performing (R_4)−b*(R_2)→(R_4)

[[1,0,1,−1],[0,1,2,−4],[0,0,0,0],[0,0,c−a−2*b,3+a+4*b]]

6. Swap the third and fourth rows to move the zero row to the bottom.

[[1,0,1,−1],[0,1,2,−4],[0,0,c−a−2*b,3+a+4*b],[0,0,0,0]]

7. Normalize the third row by dividing by c−a−2*b (assuming c−a−2*b≠0.

[[1,0,1,−1],[0,1,2,−4],[0,0,1,(3+a+4*b)/(c−a−2*b)],[0,0,0,0]]

8. Eliminate the third entry in the first row by performing (R_1)−(R_3)→(R_1)

[[1,0,0,−1−(3+a+4*b)/(c−a−2*b)],[0,1,2,−4],[0,0,1,(3+a+4*b)/(c−a−2*b)],[0,0,0,0]]

9. Eliminate the third entry in the second row by performing (R_2)−2*(R_3)→(R_2)

[[1,0,0,(a+2*b−c−3−a−4*b)/(c−a−2*b)],[0,1,0,−4−(6+2*a+8*b)/(c−a−2*b)],[0,0,1,(3+a+4*b)/(c−a−2*b)],[0,0,0,0]]

10. Simplify the constants in the fourth column.

[[1,0,0,(−2*b−c−3)/(c−a−2*b)],[0,1,0,(−4*c+4*a+8*b−6−2*a−8*b)/(c−a−2*b)],[0,0,1,(3+a+4*b)/(c−a−2*b)],[0,0,0,0]]

Final Answer

RREF*[[1,0,1,−1],[2,−1,0,2],[0,1,2,−4],[a,b,c,3]]=[[1,0,0,(2*b+c+3)/(a+2*b−c)],[0,1,0,(2*a−4*c−6)/(c−a−2*b)],[0,0,1,(a+4*b+3)/(c−a−2*b)],[0,0,0,0]]

Want more problems? Check here!