Answer
See answer below
Work Step by Step
Let $S$ be the set set of $2 \times 2$ real matrices whose entries are either all zero or all nonzero.
We have $A=\begin{bmatrix}
-1& 1\\
1 & 1
\end{bmatrix}$ and $A=\begin{bmatrix}
1& 1\\
1 & 1
\end{bmatrix}$ are all in $S$.
But if we take $A+B=\begin{bmatrix}
-1& 1\\
1 & 1
\end{bmatrix}+\begin{bmatrix}
1& 1\\
1 & 1
\end{bmatrix}=\begin{bmatrix}
0 & 2\\
2 & 2
\end{bmatrix}$ then $A+B$ is not in $S$.
Hence $S$ do not form a vector spcae over $R$
