TITLE: Interesting subspace of $M_n(\mathbb{C})$ [CMI 2019]
QUESTION [5 upvotes]: Cosider the vector space $V=M_n(\mathbb C)$ and $W$ be a vector subspace of $V$ such that every non-zero element of $W$ is invertible. Show that dim$W=1$. 
I know that any matrix over $\mathbb C$ is triangulable. I assumed that dim$W$ is 2, Can I show that this gives rise to a singular matrix.

REPLY [5 votes]: Let $A,B\in W$ be two linearly independent matrix, then $A^{-1}B$ is invertible (not necessarily to be in $W$). Now $A^{-1}B$ has an eigenvector $v$, corresponding to eigenvalue $\lambda\in \mathbb{C}$ (the field $\mathbb{C}$ implies the existence of eigenvalue). We have $A^{-1}Bv = \lambda v$. This implies $(B-\lambda A)v = 0$ and so $B-\lambda A$ is not invertible.