# Definition:Simultaneous Linear Equations/Matrix Representation

## Definition

A system of simultaneous linear equations can be expressed as:

$\mathbf A \mathbf x = \mathbf b$

where:

$\mathbf A = \begin {bmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {bmatrix}$, $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf b = \begin {bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {bmatrix}$

are matrices.

### Matrix of Coefficients

The matrix $\mathbf A$ is known as the matrix of coeffficients of the system.

### Augmented Matrix

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.