\begin{definition}[Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix]
Consider the '''system of simultaneous linear equations''' can be expressed as:
:$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
expressed in matrix representation as:
:$\mathbf A \mathbf x = \mathbf b$
Let $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ be the block matrix formed from $\mathbf A$ and $\mathbf b$.
Then $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ is known as the '''augmented matrix''' of the system.
Thus:
:$\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix} = \begin {bmatrix}
\alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} & \beta_1 \\
\alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} & \beta_2 \\
      \vdots &       \vdots & \ddots & \vdots & \vdots \\
\alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} & \beta_m \\
\end {bmatrix}$
\end{definition}