Since scalar multiplication remains unchanged, we only need to consider the axioms that involve vector addition. Since every sum is the zero matrix, the first 4 properties hold easily. However, there is no zero vector in this set. Suppose that there was. Then there is a matrix $Z$ so that $A + Z = A$ for any $2\times 2$ matrix $A$. However, $A + Z = \zeromatrix_{2,2}$, which is in general not equal to $A$, so Property Z fails and this set is not a vector space.