The final matrix that we row-reduced in Example CSANS should look familiar in most respects to the procedure we used to compute the inverse of a nonsingular matrix, Theorem CINM. We will now generalize that procedure to matrices that are not necessarily nonsingular, or even square. First a definition.
Suppose $A$ is an $m\times n$ matrix. Extend $A$ on its right side with the addition of an $m\times m$ identity matrix to form an $m\times (n + m)$ matrix $M$. Use row operations to bring $M$ to reduced row-echelon form and call the result $N$. $N$ is the extended reduced row-echelon form of $A$, and we will standardize on names for five submatrices ($B$, $C$, $J$, $K$, $L$) of $N$.
Let $B$ denote the $m\times n$ matrix formed from the first $n$ columns of $N$ and let $J$ denote the $m\times m$ matrix formed from the last $m$ columns of $N$. Suppose that $B$ has $r$ nonzero rows. Further partition $N$ by letting $C$ denote the $r\times n$ matrix formed from all of the nonzero rows of $B$. Let $K$ be the $r\times m$ matrix formed from the first $r$ rows of $J$, while $L$ will be the $(m-r)\times m$ matrix formed from the bottom $m-r$ rows of $J$. Pictorially, \begin{equation*} M=[A\vert I_m] \rref N=[B\vert J] = \left[\begin{array}{c|c}C&K\\\hline0&L\end{array}\right] \end{equation*}
Suppose that $A$ is an $m\times n$ matrix and that $N$ is its extended echelon form. Then
Notice that in the case where $A$ is a nonsingular matrix we know that the reduced row-echelon form of $A$ is the identity matrix (Theorem NMRRI), so $B=I_n$. Then the second conclusion above says $JA=B=I_n$, so $J$ is the inverse of $A$. Thus this theorem generalizes Theorem CINM, though the result is a “left-inverse” of $A$ rather than a “right-inverse.”
The third conclusion of Theorem PEEF is the most telling. It says that $\vect{x}$ is a solution to the linear system $\linearsystem{A}{\vect{y}}$ if and only if $\vect{x}$ is a solution to the linear system $\linearsystem{B}{J\vect{y}}$. Or said differently, if we row-reduce the augmented matrix $\augmented{A}{\vect{y}}$ we will get the augmented matrix $\augmented{B}{J\vect{y}}$. The matrix $J$ tracks the cumulative effect of the row operations that converts $A$ to reduced row-echelon form, here effectively applying them to the vector of constants in a system of equations having $A$ as a coefficient matrix. When $A$ row-reduces to a matrix with zero rows, then $J\vect{y}$ should also have zero entries in the same rows if the system is to be consistent.