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First, one refers to the entries of a matrix, and not the elements of a matrix since a matrix is not a set — this is the crux of the issue, as the symbol $`\in'$ is only appropriate in the context of sets.
One way to resolve this is the following: if $A= [a_{ij}]$ is an $m$-by-$n$ matrix with complex entries, one could define the elements of $A$ as the set $E(A):=\{ x \in \mathbb{C}\mid x = a_{ij},1\le i \le m, 1\le j \le n\}$. For example, if $$ A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, $$ then $E(A) = \{ a_{11},a_{12},a_{21},a_{22}\}$. In this setting it is appropriate to write $`a_{ij} \in E(A)'$.
No, in a matrix $$A = \begin{bmatrix}a&b\\c&d\end{bmatrix},$$ the "entries" $a,b,c,d$ are not "elements" of $A$.
Similarly, in an ordered triple $$ \mathbf{x} = (a,b,c) $$ the "coordinates" $a,b,c$ are not "elements" of $\mathbf{x}$.
In a string $$ \sigma = \text{"help"} $$ the "letters" h,e,l,p are not "elements" of $\sigma$.