In Gauss-Jordan elimination, we convert the augmented matrix to reduced row echelon form (RREF). RREF is very similar to row echelon form, where the leading entry of each row is 1, and the column containing a leading 1 has all other entries as 0.
A matrix that can be obtained from the identity matrix, by applying elementary row and column transformations, is known as an elementary matrix.
Interestingly, premultiplying a matrix by an elementary matrix which was obtained by applying a certain set of elementary transformations on the identity matrix, is the same as applying those transformations
Fundamental Theorems of Invertible Matrices:
a) $A$ is invertible.
b) $Ax = B$ has a unique solution for every $b \in \mathbb{R}^n$.
c) $Ax = 0$ has only the trivial solution.
d) The RREF of $A$ is $I_n$.
e) $A$ is a product of elementary matrices.
The above statements are all equivalent forms of each other.
Definition: (Invertible Matrices)
If $A$ is an $n \times n$ matrix, the inverse of $A$ is a matrix $A^{-1}$ with the property
\(AA^{-1} = A^{-1}A = I_n\)
If $Ax = B$ has a unique solution, $x = A^{-1}B$.
Suppose there exists another solution $y$, then
\(Ay = B\)
\(\implies A^{-1}Ay = A^{-1}B\)
\(\implies Iy = A^{-1}B\)
\(\implies y = A^{-1}B\)
Thus we have $y = x = A^{-1}B$, meaning we only have a unique solution.