\begin{definition}[Definition:Equality of Matrices]
Let $\mathbf A$ and $\mathbf B$ be matrices over an underlying structure $R$.
Then $\mathbf A$ is equal to $\mathbf B$ {{iff}}:
:$(1): \quad$ the order of $\mathbf A$ equals the order of $\mathbf B$: $m \times n$, say
:$(2): \quad$ for all $i \in \set {1, 2, \ldots, m}$ and $j \in \set {1, 2, \ldots, n}$, we have that $a_{i j} = b_{i j}$.
\end{definition}