Determine all possible RREF $$a_1x+b_1y+c_1z=0\\
a_2x+b_2y+c_2z=0\\
a_3x+b_3y+c_3z=0$$
I'm new to Linear Algebra. But from what i understand isn't there only one possible RREF:
$$\left(\begin{array}{ccc|c}
1&0&0&0\\
0&1&0&0\\
0&0&1&0
\end{array}\right)$$
 A: Not really, on some conditions you may get RREF with only two or even one pivot, which depends on the entries of coefficient matrix.
A: Note that since the RHS is homogenous, it will always be $0$. Also note that we can make all non-diagonal elements $0$ if there is a non-zero pivot, so any RREF will be of the form
$$\left(\begin{array}{ccc|c}
\ast&.&.&0\\
0&\ast&.&0\\
0&0&\ast&0
\end{array}\right)$$
Now if a diagonal element is non-zero, we can scale it to $1$. If it is zero however, we can only swap rows to move it. This means the diagonal is defined by the number of $1$s in it (This is called the rank of the matrix). Therefor all possible RREFs are
$$\left(\begin{array}{ccc|c}
1&0&0&0\\
0&1&0&0\\
0&0&1&0
\end{array}\right)\\
\left(\begin{array}{ccc|c}
1&0&.&0\\
0&1&.&0\\
0&0&0&0
\end{array}\right)\\
\left(\begin{array}{ccc|c}
1&.&.&0\\
0&0&0&0\\
0&0&0&0
\end{array}\right)\\
\left(\begin{array}{ccc|c}
0&0&0&0\\
0&0&0&0\\
0&0&0&0
\end{array}\right)$$
Note that the last is only possible if the original matrix is already the zero-matrix. The $.$ can be any real number.
