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Let \( G = G(v, e) \) be a finite simple graph with \( v \) vertices and \( e \) edges. An \((a, d)\)-\text{edge-antimagic-vertex} (EAV) \text{labeling} is a one-to-one mapping \( f: V(G) \to \{1, 2, \ldots, v\} \) such that for every edge \( xy \in E(G) \), the edge-weight set
\[
\{f(x) + f(y) \mid xy \in E(G)\} = \{a, a+d, a+2d, \ldots, a+(e-1)d\}
\]
for some positive integers \( a \) and \( d \). An \((a, d)\)-\text{edge-antimagic-total labeling} is a one-to-one mapping \( f: V(G) \cup E(G) \to \{1, 2, \ldots, v+e\} \) with the property that for every edge \( xy \in E(G) \),
\[
\{f(x) + f(y) + f(xy) \mid xy \in E(G)\} = \{a, a+d, a+2d, \ldots, a+(e-1)d\}.
\]
This labeling is called \text{super \((a, d)\)-edge-antimagic total labeling} if \( f(V(G)) = \{1, 2, \ldots, v\} \). In this paper, we investigate the relationship between the adjacency matrix, \((a, d)\)-edge-antimagic vertex labeling, and super \((a, d)\)-edge-antimagic total labeling, and show how to manipulate this matrix to construct new \((a, d)\)-edge-antimagic vertex labelings and new super \((a, d)\)-edge-antimagic total graphs.