Answer
See below
Work Step by Step
We can write set $S$ as $S=\{A \in M_n(R)\}$
We can notice that $n \times n$ matrix $O$ such as $o_{ij}=0$ for all $1 \leq i \leq n$ and $1 \leq j \leq n$ is in $S$. Hence, $S$ is nonemtpy.
Let $A,B \in S$. We have $a_{ij}=0$ and $b_{ij}=0$. Since $i \lt j \rightarrow a_{ij}+b_{ij}=9 \rightarrow A+B \in S$
Let $A \in S$ and a scakar $c$. We have $a_{ij}=0$. Since $i \lt j \rightarrow ca_{ij}=c0=0 \rightarrow cA \in S$
Hence, $S$ is a nonempty subset of $M_n(R)$.
Thus, $S$ is a subspace of $M_n(R)$
