Here is how you can perform row operations on a given matrix.
Let us start by creating an augmented matrix for the following system of equations:
 
x + 4 y + 3 z = 10,
2 x + y - z = -1,
3 x - y - 4 z = 11
 
 Here it is (the semicolon supresses the visible output from the assignment command):

M=(({{1, 4, 3, 10}, {2, 1, -1, -1}, {3, -1, -4, 11}}));
MatrixForm[M]

To gain control over the individual rows of M, we use the double square brackets: for example M[[2]] indicates the second row of M. The first code produces row exchange in a matrix M.

Temp=M;
M[[1]]=Temp[[2]];
M[[2]]=Temp[[1]];
MatrixForm[M]

The following code subtracts twice the first row from the second.

M[[2]]=M[[2]]-2*M[[1]];
MatrixForm[M]