I'm starting out working with RREF and I wanted to work through an example problem. One was to compute the reduced row echelon form of the following matrix:
$$\begin{bmatrix}-1 & 1\\-1 & 2\\-3 & 2\end{bmatrix}$$
I'm a bit confused, because if I'm understanding correctly, this is supposed to be an augmented matrix (so I can technically draw a line down the middle in this case to make it clearer).
But that would mean that it corresponds to a system as follows:
$$-x_1=1$$ $$-x_1=2$$ $$-3x_1=2$$
And when I did the reduction, I got:
$$\begin{bmatrix}1 & 0\\0 & 1\\0 & 0\end{bmatrix}$$
So that would correspond with $x_1 =0$ but $0 = 1$ so from my understanding, the solution is inconsistent? I'm wondering if a) I interpreted the augmented matrix correctly and the corresponding system aligns with the matrix, and b) if the fact that row 2 is $0\ \ \ \ 1$ means that it's inconsistent?
Yes, the RREF is indeed correct.
$$0=1$$ implies that the system is inconsistent.
Remark: to further convince yourself that it is indeed inconsistent. The first equation says that $x_1=-1$ btu the second equation says that $x_1=-2$. $-1=-2$ is a contradiction.