Answer
See below
Work Step by Step
We can write set $S$ as $S=\{A =\begin{bmatrix}
a & b & c \\ d & e &f
\end{bmatrix} \in M_{2\times 3}(R): a+b+c=10,d+e+f=10\}$.
We can notice that $S=\{A =\begin{bmatrix}
0 & 0 & 0 \\ 0 & 0 &0
\end{bmatrix} \notin S$,$S$ is not a subspace of $M_{2\times 3}(R)$
