\begin{definition}[Definition:Simultaneous Linear Equations/Matrix Representation]
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.
\end{definition}