linear algebra - Reduced Row Echelon Form of 3 x 4

I have the matrix $$\pmatrix{-1 & -2 & 2 & -12 \\ 4 & 1 & -3 & 17 \\ 2 & -3 & 3 & -11}$$

I am trying to bring this to reduced row echelon form. Here are the operations I performed: $$R_2-2R_3 \\ R_3 + 2R_1 \\ R_3 + R_2 \\ - R_1 \\ R_2/7 \\ R_3/-2 \\ R_1-2R_2 \\ R_2+ (9/7) R_3$$ I arrived at $$\pmatrix{1 & 0 & 4/7 & 6/7 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -2}$$ I know the normal way to proceed is to create a leading 1 for the first row and then add and subtract multiples of the first row from other rows but why wont the above operations give me a correct RREF? Is there an inherent problem in the above operations? If so, what is it?

Answer

You're almost done you just have to do R1 - (4/7)R3 to get rid of the 4/7 in R1.