I am not sure if my answer is correct.
If $$S=\left\{\begin{pmatrix} a &b \\ c & d \end{pmatrix}\;\middle\vert\; ad=0 \quad a,b,c,d\in \mathbb{R} \right\}$$ it means that or $a=0$ or $d=0$ or both of them equals $0$. Then is $$\mathrm{span}(\bigl(\begin{smallmatrix} a&b \\ c& 0 \end{smallmatrix}\bigr),\bigl(\begin{smallmatrix} 0 &b \\ c & d \end{smallmatrix}\bigr))= S$$
If not so, what is $k$ so $\mathrm{span}(k)=s$?
We know that $V=M_2(\mathbb{R})$ is a $4$-dimensional real vector space. Hence $U:={\rm span}{S}$ is a subspace of dimension $r\le 4$. Note that $S$ is not a linear subspace. We clearly have $r\ge 4$, since every matrix in $V$ is a linear combination of matrices in $S$. Hence $\dim U=4$, so that the span of $S$ is equal to $M_2(\mathbb{R})$.
Edit: The answer refers to the original question, which is a well known question. The new one is less interesting and has a trivial asnwer.