Definition 1: Let $A={[a_{ij}]}_{m\times n}$ and $B={[b_{ij}]}_{m\times n}$. We define $$A+B:={[a_{ij}+b_{ij}]}_{m\times n}$$
My problem: Which of the following mathematical notations below is correct for the definition provided above?
$$ \begin{array}{ll} (1) & (A={[a_{ij}]}_{m\times n}\wedge B={[b_{ij}]}_{m\times n})\Longleftrightarrow A+B={[a_{ij}+b_{ij}]}_{m\times n}\\ (2) & (A={[a_{ij}]}_{m\times n}\wedge B={[b_{ij}]}_{m\times n})\Longrightarrow A+B={[a_{ij}+b_{ij}]}_{m\times n}\\ (3) & A={[a_{ij}]}_{m\times n}\wedge B={[b_{ij}]}_{m\times n} \wedge A+B={[a_{ij}+b_{ij}]}_{m\times n}\\ \end{array} $$
Definition 2: How can I express the following definition using logical symbols?
If $A={[a_{ij}]}_{m\times n}$ and $B={[b_{ij}]}_{p\times q}$ then
$$ A=B\Longleftrightarrow m=p,\quad n=q,\quad a_{ij}=b_{ij}\text{ for all }i,j $$
What I think: When computing $A + B$, I verify if both are matrices, suggesting that option (3) might be correct. However, I've heard that definitions are often biconditional. Yet, option $(2)$ appears more plausible, as the reverse might not be uniquely determined.
It is important to stress that your Definition 1 is perfectly good mathematical notation as it stands.
Sprinkling around logical notation here does nothing helpful. And in fact the correct formula (2) is less informative than the ordinary mathematic presentation which explicitly says that a definition is being presented. A conditional proposition like (2) of the form $A \Rightarrow B$ may or may not be true as a matter of definition.
Logical notation is great in its place when it can do clarificatory work: but this is not the place.