## Elementary Linear Algebra 7th Edition

$W$ is not a vector subspace of $M_{n,n}$.
Let $u\in W$ and $c=2\in R$. Now, $$\det(u)=1, \quad \det(cu)=c\det(u)=2$$ which means that $cu$ is not an element of $W$. Hence, $W$ is not a vector subspace of $M_{n,n}$.