augmented matrix

An augmented matrix $(A\mid B)$ is a $m\times (n+k)$ matrix constructed by appending a $m\times k$ matrix $B$ to a $m\times n$ matrix $A$.

This is typically done when solving a linear system, for the purposes of performing Gaussian elimination.

Given a matrix equation $A\overrightarrow{x}=\overrightarrow{b}$, the augmented matrix $(A\mid \overrightarrow{b})$ can be reduced to row echelon form, which gives information about the consistency of the system.

For a matrix $A$ and identity matrix $I$, the augmented matrix $(A\mid I)$ can be reduced using elementary row operations to $(I\mid A^{-1})$, giving the inverse of $A$.