$ V = M_{2,2} (\mathbb{R}), \forall M, N \in V : M \oplus N = MN, \forall M \in V, c \in \mathbb{R} : c \odot M = cM $
I know $\oplus$ and $\odot$ are closed, but a vectorspace also needs a neutral element, which would be $I_{2}$ in this case, and an inverse element which would be for $M^{-1}$ for $M \in V$. Now the inverse of M is not necessarily defined on a matrix, so would that be enough to disqualify this as a vectorspace? Would I need to take $V = Gl_2(\mathbb{R})$ for this to be a vectorspace? But then $\oplus$ might not be closed?