Answer
See below
Work Step by Step
We can write set $S$ as $S=\{A \in M_{2}(R)\}$ and $A$ is symmetric.
We can notice that $S=\{A =\begin{bmatrix}
0 & 0 \\ 0 &0
\end{bmatrix} \in S$, hence $S$ is nonempty.(1)
Let $A,B \in S$
Obtain $A^T=A, B^T=B$ then $(A+B)^T=A^T+B^T=A+B \rightarrow A+B \in S$
Hence, $A+B$ is closed under addition. (2)
Then let $A\in S$ and $c$ be a scalar.
Obtain $(cA)^T=cA^T=cA \rightarrow cA \in S$
Hence, $cA$ is closed under scalar multiplication (3)
From (1),(2),(3), $S$ is a subspace of $M_{2}(R)$
