Linear System Theorems

Consistency Criterion

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column.

In other terms, if there exists a row in the augmented matrix $[0,0,0,C]$ where $C$ is a nonzero constant, the system is inconsistent.

Infinitude of Solutions

If the system is consistent and there are no free variables, there is exactly one solution. If there are one or more free variables, there are an infinite number of solutions.

This can only be applied after first confirming the system is consistent.