Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix

Definition
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}$