Let's have a matroid built from a matrix and a matroid built from a graph and construct the new object and see if we encounter anything bothering us because of the difference of motivation of the two matroids or not.
Take $A$, a $2\times 2$ matrix as $\begin{bmatrix} 1 & 0\\ 0 & 0\end{bmatrix}$ and Define the Graph $G$ be the complete graph on 3 vertices, meaning $K_3$ with edges called $a$, $b$ and $c$. Then if I call the matroids coming from them with $M_1$ and $M_2$ respectively, we have (index columns of $A$ with $1$ and $2$ from left to right);
$\begin{array}{l} E(M_1)=\{1,2\}\\ I(M_1)=\{\emptyset,\{1\}\}\\ E(M_2)=\{a,b,c\}\\ I(M_2)=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\} \end{array}$
Conditions of your sum theorem are hold.
$\begin{array}{l} E(M):=\{1,2,a,b,c\}\\ I(M)=\{\emptyset,\{1\},\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\} \end{array}$
As you can check readably, this is indeed a matroid.