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?
