TITLE: Problems concerning subspaces of $M_n(\mathbb{C})$
QUESTION [19 upvotes]: Let $M_n(\mathbb{C})$ denote the n times n matrices over the complex number field. N be a subspace of $M_n(\mathbb{C})$.

If all the matrices in N are non-invertible , what is the maximum the dimension of N can be?
If all the matrices in N commute with each other, what is the maximum the dimension of N can be?
If all the matrices in N are nilpotent, what is the maximum the dimension of N can be?
If all the non-zero matrices in N are invertible, what is the maximum the dimension of N can be?

REPLY [6 votes]: For problem 4 over the real field, the answer is the Radon-Hurwitz function at $n$. See for instance Petrovic, "On nonsingular matrices and Bott periodicity." The Radon-Hurwitz function is defined to be $\rho(n)=8a+2^b$, where the largest power of 2 dividing $n$ is $2^{4a+b}$, $a\geq 0$, $0\leq b\leq 3$.