I know that inconsistency is detected by looking at the rref of the augmented matrix, but let's say that i have the rref of the coefficient matrix.
AKA I have the coefficients of the variables, but none of the constants.
(or alternately, i have the kernel or null space, if that's easier)
How would I go about knowing whether inconsistency was possible or not in that system of equations?
I can reason about simple examples but is there a formal way to see whether inconsistency is possible, or details of how inconsistency could be made?
Some examples I can reason about below.
These equations can't be inconsistent:
$x+y=A\\ y=B\\ A,B \in \mathbb{R} $
They have this coefficient matrix in rref:
$ \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} $
These can be inconsistent, but aren't necessarily:
$x+y=A\\ x=B\\ y=C \\ A,B,C \in \mathbb{R} $
And here is the coefficients matrix in rref for those:
$ \begin{bmatrix} 1 & 0\\ 0 & 1\\ 0 & 0 \end{bmatrix} $
You cannot. Let's say you have a nonhomogeneous system of two equations in two variables and I tell you that the rref form of the system is
$$ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}. $$
From this information you can deduce that the original system doesn't have a unique solution for each $(A,B)$ you put in the right hand side and when there are solutions, there will be infinitely many but you cannot tell for a specific $(A,B)$ (say $(1,2)$) whether the system will have a solution.
To see an example, consider the systems $x = A, 0 = B$ and $x = A, x = B$. The rref of the coefficient matrices
$$ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} $$
is the same but the first system can be solved for $(1,0)$ while the second cannot be solved for $(1,0)$.