To track the effect of a matrix (here a elimination matrix as an example) on the entire system in instead of just the matrix operation , we can augment as an additional column into the matrix . We go from this structure, in which we need to separately calculate and
E \underbrace{ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} }_{ \text{Matrix} A } \= E \underbrace{ \begin{bmatrix} 2 \\ 2 \\ 2 \\ \end{bmatrix} }_{ \text{Matrix} B } \hspace{1cm} \text{( $x$ was neglected here )} $$toE \begin{bmatrix} 1 & 1 & 1 & 2\ 1 & 1 & 1 & 2\ 1 & 1 & 1 & 2\ \end{bmatrix}
We appended $b$ as additional column of $A$. Augmentation is possible through the [[Associative Law for Matrices#splitting-colums|Associative Law]] for matrices. Whatever we now calculate as a result, the last column of our $3 \times 4$ matrix will be the vector $b$ equally transformed by $E$ as $A$ is.