So, if we have a matrix $A$ and an augmented matrix $A\mid n$, if the $\operatorname{rank(A)}\ne\operatorname{rank}(A|b)$, does that mean that the system is inconsistent?
I'm trying to understand the cases for unique solutions, an infinite number of solutions, and an inconsistent system in relation to rank of that system. Thanks! :)
An augmented matrix is a representation of a Linear Equations System so if $A$ is the coefficients matrix and $A|b$ is the aumented matrix of the System, $\mathrm{rank}\,A \neq \mathrm{rank}\,A|b$ means that $A|b$ has a pivot that $A$ has not, that leads to a contradiction of the type $ 0 = a$ in the system, where $a$ is a non-zero constant. You can verify that if $A$ is an $m\times n$ matrix and $\mathrm{rank}\,A =\mathrm{rank}\,A|b = n$ then the solution is unique; on the other hand if $\mathrm{rank}\,A =\mathrm{rank}\,A|b < n$ then the system has infinitely many solutions.