Chapter 4 - Vector Spaces - 4.3 Subspaces of Vector Spaces - 4.3 Exercises - Page 167: 18

$W$ is not a vector subspace of $M_{n,n}$.

Let $A, B$ be two $n\times n$ matrices in $W$. Then, $A^2=A$ and $B^2=B$. Now, $$(A+B)^2=A^2+B^2+AB+BA\\ \hspace{1.4cm}=A+B+AB+BA$$ which means that $A+B$ does not belong to $W$. Hence, $W$ is not a vector subspace of $M_{n,n}$.