A matrix is a collection of numbers that can be indexed in a particular way, this indexing mimics a two dimensional grid. The dimensions of a matrix are stated as \( r \times c \) where \( r, c \in \mathbb{ N } _ 1 \) when it has \( m \) rows and \( n \) columns.
Suppose that \( A \) is a matrix, then we can access the value at the \( i \)-th row and \( j \)-th column by writing \( A_{i , j} \), we can graphically represent this information by introducing variables for the coefficients and writing them like this:
\[ A = \begin{bmatrix} a_{1 , 1} & a_{1 , 2} & \ldots & a_{1 , n} \\ a_{2 , 1} & a_{2 , 2} & \ldots & a_{2 , n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m , 1} & a_{m , 2} & \ldots & a_{m , n} \end{bmatrix} \]