Definition of extended reduced row echelon form of a matrix

Definition: The extended reduced row echelon form of an $m \times n$ matrix $A$ is obtained by augmenting $A$ with the $m \times m$ identity matrix and row-reducing:

$$[A \mid I_m] \xrightarrow{\text{row reduce}} [R \mid E]$$

where $R$ is the RREF of $A$ and $E$ is the product of elementary matrices that performed the reduction.

Interpretation: The matrix $E$ records the cumulative effect of all row operations. If $A$ is square and invertible, then $R = I$ and $E = A^{-1}$. For general matrices, $E$ provides information about the null space and left null space of $A$.