Theorem 1 - Uniqueness of the Reduced Echelon Form
Each matrix is row equivalent to exactly one reduced echelon matrix.
Theorem 2 - Existence and Uniqueness Theorem
A linear system is consistent if and only if the rightmost column of the augmented matrix is pivot column - that is, if and only if an echelon form of the augmented matrix has no row of the form(1)
with b nonzero
If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.